Geometry worksheet with angle puzzles involving intersecting lines and algebraic expressions for angle measures.

Let’s solve this step by step.

We are given two diagrams (A and B) with angles labeled m∠1 through m∠20, and some angle measures provided. We need to find all missing angle measures using geometry rules: vertical angles, linear pairs, corresponding angles, alternate interior/exterior angles, etc., especially since lines appear parallel in both diagrams.

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Diagram A:

Given:
- m∠7 = 45°
- m∠14 = 85°

Lines a and b are horizontal and likely parallel (since they’re cut by transversals c and d). Lines c and d are transversals crossing them.

Let’s label the intersections:

Top line: points 1–6 on line a
Bottom line: points 13–18 on line b
Transversal c goes from top-left to bottom-right (through 1, ∠5, ∠9, ∠13)
Transversal d goes from top-right to bottom-left (through ∠2, ∠6, ∠10, ∠14)

Wait — actually looking at numbering:

In Diagram A:

At top intersection (line a and transversal c): angles 1, 2, 3, 4
At top intersection (line a and transversal d): angles 5, 6, 7, 8? Wait no — let's map carefully.

Actually, standard labeling for such puzzles:

Usually, each intersection has 4 angles numbered consecutively around the point.

Looking at diagram A:

There are two transversals cutting two parallel lines → 4 intersection points.

But the numbers go up to 14? Actually, let’s count:

Angles listed: m∠1 to m∠14 → so 14 angles total.

From the drawing description (even though we can’t see it), typical setup:

Line a (top horizontal) intersected by transversal c (left diagonal) → creates angles 1,2,3,4
Same line a intersected by transversal d (right diagonal) → creates angles 5,6,7,8
Line b (bottom horizontal) intersected by transversal c → angles 9,10,11,12
Line b intersected by transversal d → angles 13,14,15,16? But only up to 14 is shown? Wait, problem says “m1 = ___ ... m∠14 = ___” and gives m∠7=45°, m∠14=85°.

Also note: In diagram A, there’s a triangle formed between the two transversals and the bottom line? Angles 9,10,13,14 are involved.

Actually, let’s use logic based on standard angle relationships.

Assume lines a and b are parallel (as is typical in these worksheets unless stated otherwise).

Given: m∠7 = 45°, m∠14 = 85°

First, identify where ∠7 and ∠14 are.

Typically:

At top right intersection (line a and transversal d):

If ∠7 is one of the angles there, and assuming standard position:

Suppose at each intersection, angles are numbered clockwise starting from top-left.

But without image, we must rely on common patterns.

Alternative approach: Use the fact that in many such problems, angles across from each other (vertical) are equal, adjacent on straight line sum to 180°, and if lines are parallel, corresponding/alternate angles are equal.

Also, notice that in diagram A, angles 9, 10, and maybe 13 or 14 form a triangle? The problem might imply that.

Wait — look at the given: m∠7 = 45°, m∠14 = 85°

And we have to find others.

Let me try to reconstruct logically.

Assume:

- Line a || line b
- Transversal c cuts them → creates angles 1,2,3,4 on top; 9,10,11,12 on bottom
- Transversal d cuts them → creates angles 5,6,7,8 on top; 13,14,15,16 on bottom — but only up to 14 is asked? Problem lists up to m∠14.

Actually, the worksheet shows:

For Diagram A:

Find m∠1 to m∠14

Given: m∠7 = 45°, m∠14 = 85°

Now, let’s assume standard positions:

At top-left intersection (a and c):

∠1 (top-left), ∠2 (top-right), ∠3 (bottom-right), ∠4 (bottom-left)

At top-right intersection (a and d):

∠5 (top-left), ∠6 (top-right), ∠7 (bottom-right), ∠8 (bottom-left)

At bottom-left intersection (b and c):

∠9 (top-left), ∠10 (top-right), ∠11 (bottom-right), ∠12 (bottom-left)

At bottom-right intersection (b and d):

∠13 (top-left), ∠14 (top-right), ∠15 (bottom-right), ∠16 (bottom-left) — but we only need up to 14.

Given m∠7 = 45° → that’s at top-right intersection, bottom-right angle.

So at that intersection (a and d), 7 = 45°

Then its vertical angle would be ∠5? No — vertical to ∠7 is ∠5 if numbered properly? Let’s think.

If at an intersection, angles are:

Top-left: ∠5
Top-right: ∠6
Bottom-right: ∠7
Bottom-left: ∠8

Then vertical angles: ∠5 and ∠7 are NOT vertical — ∠5 and ∠7 are opposite? Actually, vertical angles are across from each other.

Standard: at intersection, vertical pairs are (5, ∠7) and (∠6, ∠8)? No.

Actually, if you have four angles around a point:

Label them as:

Northwest: ∠5
Northeast: ∠6
Southeast: ∠7
Southwest: ∠8

Then vertical angles are: ∠5 and ∠7 (diagonally opposite), ∠6 and 8.

Yes! So if ∠7 = 45°, then its vertical angle ∠5 = 45°.

Also, adjacent angles on straight line: ∠5 + ∠6 = 180°, so ∠6 = 180 - 45 = 135°

Similarly, ∠7 + ∠8 = 180° → ∠8 = 135°

Now, since line a || line b, and transversal d cuts them, then corresponding angles should be equal.

∠7 is at top-right intersection, southeast corner.

Corresponding angle on bottom line would be at bottom-right intersection, southeast corner — which would be ∠15? But we don’t have 15 in our list.

Wait, we have ∠14 given as 85°.

∠14 is at bottom-right intersection, northeast corner? If:

At bottom-right intersection (b and d):

∠13 (northwest), ∠14 (northeast), ∠15 (southeast), ∠16 (southwest)

Given m∠14 = 85°

Then vertical angle to ∠14 is ∠16 = 85°

Adjacent: ∠13 + ∠14 = 180° → ∠13 = 95°

∠14 + ∠15 = 180° → ∠15 = 95°

Now, back to parallel lines.

Transversal d: angles on top line a: ∠5, 6, ∠7, ∠8

On bottom line b: ∠13, ∠14, ∠15, 16

Corresponding angles:

∠5 corresponds to ∠13? Let's see positions.

If transversal d goes from top-right to bottom-left, then:

At top: ∠5 is northwest, ∠6 northeast, ∠7 southeast, ∠8 southwest

At bottom: since line b is below, and transversal comes from top-right to bottom-left, then:

∠13 is northwest (same side as ∠5), ∠14 northeast, etc.

Actually, for corresponding angles, same relative position.

So ∠5 (top, left of transversal, above line) corresponds to ∠13 (bottom, left of transversal, above line) — yes.

Since lines are parallel, corresponding angles equal → ∠5 = ∠13

But we have ∠5 = 45° (from vertical to ∠7), and ∠13 = 95° (from adjacent to ∠14=85°) — contradiction!

That means my assumption about which angle is which is wrong.

Perhaps ∠7 is not at the top-right intersection.

Maybe the numbering is different.

Another possibility: in some worksheets, the angles are numbered sequentially along the transversals.

Look at the given: m∠7 = 45°, m∠14 = 85°

And in diagram A, there is a triangle formed by the two transversals and the bottom line, with angles at 9, 10, and 13 or 14.

Notice that angles 9, 10, and the angle between the transversals at the bottom might form a triangle.

Specifically, at the bottom, between the two transversals, there is a triangle with vertices at the intersections.

For example, the triangle might have angles at 9, ∠10, and the angle at the vertex where the two transversals meet the bottom line — but they meet at different points.

Actually, the two transversals cross each other somewhere, forming a triangle with the bottom line.

In diagram A, likely the two transversals intersect between the two parallel lines, forming a triangle with the bottom line segment between the two intersection points.

So, the triangle has three angles: one at left intersection on bottom line (say ∠9 or ∠10), one at right intersection on bottom line (∠13 or ∠14), and one at the intersection point of the two transversals.

But the angles at the transversal intersection are not labeled directly.

However, we can use the fact that the sum of angles in a triangle is 180°.

Moreover, we are given m∠7 = 45° and m∠14 = 85°.

Let me try a different strategy.

Assume that in diagram A:

- m∠7 = 45° is an angle at the top, and it is part of a pair with another angle.

Perhaps ∠7 and ∠3 are corresponding or something.

Another idea: use the fact that vertical angles are equal, and linear pairs sum to 180°, and for parallel lines, alternate interior angles are equal.

Let's start with what we know.

Given m∠7 = 45°.

Suppose ∠7 is at the top-right intersection, and it is the angle between line a and transversal d, on the lower side.

Then, the vertical angle to ∠7 is the angle directly opposite, which would be at the same intersection, say ∠5 if numbered differently.

To avoid confusion, let's define:

Let P be the intersection of line a and transversal d.

At P, the four angles are: let's call them A,B,C,D in order.

But perhaps it's better to use the given values to find related angles.

Notice that in many such problems, m∠7 and m∠3 might be related if they are corresponding or alternate.

Perhaps m∠7 and m∠11 are corresponding if lines are parallel.

Let's calculate based on the triangle.

In diagram A, there is a triangle formed by the two transversals and the bottom line. The three angles of this triangle are:

- At left bottom intersection: the angle inside the triangle, which might be ∠10 or ∠9

- At right bottom intersection: the angle inside the triangle, which might be ∠13 or ∠14

- At the intersection of the two transversals: the angle between them, which is vertically opposite to an angle at the top.

Specifically, the two transversals intersect at some point, say Q, between the two parallel lines.

At Q, the vertical angles are equal, and they are also related to the angles at the top.

For example, the angle at Q that is inside the triangle is vertically opposite to the angle at the top between the two transversals.

At the top, between the two transversals, there is an angle that is part of the angles at the top intersections.

For instance, at the top, the angle between transversal c and d on the upper side might be composed of parts of 2, ∠3, etc.

This is getting messy.

Let me look for a standard solution pattern.

I recall that in such worksheets, often m∠7 and m∠3 are vertical or corresponding.

Another approach: use the given to find adjacent angles.

Suppose m∠7 = 45°.

Then, if ∠7 and ∠8 are adjacent on a straight line, m∠8 = 180 - 45 = 135°.

Similarly, if ∠7 and ∠6 are adjacent, same thing.

But we need to know which are adjacent.

Perhaps from the diagram description, ∠7 is at the top, and it is acute, 45°, and m∠14 = 85° at the bottom.

Also, in the triangle at the bottom, the three angles are m∠9, m∠10, and the angle at the vertex where the two transversals meet the bottom line — but they meet at different points, so the triangle is formed by the segment between the two bottom intersections and the two transversals meeting at a point above.

So, the triangle has vertices at:

- Left bottom intersection (angles 9,10,11,12)

- Right bottom intersection (angles 13,14,15,16)

- And the intersection point of the two transversals, say R.

At R, the angle inside the triangle is, say, θ.

Then, the three angles of the triangle are:

- At left bottom: the angle between line b and transversal c, on the side towards the triangle. This might be ∠10 or 9.

- At right bottom: the angle between line b and transversal d, on the side towards the triangle. This might be ∠13 or ∠14.

- At R: the angle between the two transversals.

Now, the angle at R is vertically opposite to the angle at the top between the two transversals.

At the top, between the two transversals, there is an angle that is part of the angles at the top intersections.

For example, at the top, the angle between transversal c and d on the lower side might be the sum or difference of some angles.

Perhaps the angle at R is equal to m∠2 or m∠3 or something.

Let's assume that the angle at R (inside the triangle) is vertically opposite to m∠2 or m∠3.

In many diagrams, the angle between the two transversals at their intersection is vertically opposite to the angle at the top between them, which might be m∠2 + m∠3 or something, but usually it's a single angle.

Perhaps at the top, the angle between the two transversals is m∠2 if they are adjacent.

I think I need to make a reasonable assumption.

Let me assume that in diagram A:

- m∠7 = 45° is the angle at the top-right intersection, between line a and transversal d, on the lower side.

- Then, the vertical angle to ∠7 is the angle at the same intersection on the upper side, which might be ∠5, so m∠5 = 45°.

- Then, the adjacent angles: m∠6 = 180 - 45 = 135°, m∠8 = 135°.

Now, since lines a and b are parallel, and transversal d cuts them, then the corresponding angle to ∠7 on the bottom line should be equal.

∠7 is on the lower side of line a, on the right side of transversal d.

Corresponding angle on line b would be on the lower side of line b, on the right side of transversal d, which would be ∠15.

But we don't have ∠15, and we have m∠14 = 85°.

m∠14 is given, and it's at the bottom-right intersection.

If ∠14 is on the upper side of line b, on the right side of transversal d, then it is corresponding to ∠6 on the top.

Because ∠6 is on the upper side of line a, on the right side of transversal d.

So if lines are parallel, corresponding angles equal, so m∠6 = m∠14 = 85°.

But earlier I had m∠6 = 135° from m∠7 = 45°, which is contradiction.

Unless m∠7 is not adjacent to ∠6.

Perhaps at the top-right intersection, the angles are numbered differently.

Suppose at top-right intersection (a and d):

∠5 = top-left

∠6 = top-right

∠7 = bottom-right

∠8 = bottom-left

Then, ∠7 and 6 are adjacent, so if m∠7 = 45°, then m∠6 = 180 - 45 = 135°.

But if m∠14 = 85°, and if ∠14 corresponds to ∠6, then 135° = 85°, impossible.

So perhaps ∠14 corresponds to a different angle.

Maybe ∠14 corresponds to ∠8 or something.

Another idea: perhaps the lines are not parallel, but the problem doesn't state that.

In most such worksheets, the horizontal lines are parallel.

Perhaps m∠7 and m∠3 are alternate interior or something.

Let's try to use the triangle.

In diagram A, the triangle is formed by the two transversals and the bottom line, so its three angles are:

- At left bottom: the angle between line b and transversal c, which is inside the triangle. This is likely ∠10, because if ∠9 is on the other side.

Typically, for the triangle, the angles are the ones facing inward.

So at left bottom intersection, the angle of the triangle is ∠10 (between line b and transversal c, on the side towards the right).

At right bottom intersection, the angle of the triangle is ∠13 (between line b and transversal d, on the side towards the left).

At the intersection of the two transversals, the angle of the triangle is the angle between them, which is vertically opposite to the angle at the top between the two transversals.

At the top, between the two transversals, the angle on the lower side is, say, the angle between transversal c and d, which might be composed of 2 and ∠3 or something, but usually it's a single angle at the vertex.

Actually, the two transversals intersect at a point, say S, between the two parallel lines.

At S, the four angles are formed, and the one inside the triangle is, say, α.

Then, the vertical angle to α is at the top, between the two transversals, on the upper side, which might be part of the angles at the top intersections.

In particular, at the top, the angle between the two transversals on the lower side is vertically opposite to α, so it is also α.

And that angle at the top between the two transversals on the lower side is the sum of the angles from the two intersections if they are adjacent, but usually it's a single angle if the transversals cross at S.

Perhaps at the top, the angle between the two transversals is m∠2 + m∠3 or |m∠2 - m∠3|, but that's complicated.

I recall that in such problems, the angle at the intersection of the two transversals is equal to the sum or difference of the remote interior angles, but for now, let's use the given.

Let me denote:

Let T be the intersection point of the two transversals.

At T, the angle inside the triangle is β.

Then, the three angles of the triangle are:

- At left bottom: γ = m∠10 (assume)

- At right bottom: δ = m∠13 (assume)

- At T: β

Sum: γ + δ + β = 180°

Now, β is vertically opposite to the angle at the top between the two transversals on the lower side.

At the top, between the two transversals, on the lower side, that angle is, say, the angle between transversal c and d, which is part of the angles at the top.

In particular, at the top-left intersection (a and c), the angle on the lower-right side is ∠3.

At the top-right intersection (a and d), the angle on the lower-left side is ∠8.

Then, the angle between the two transversals at the top on the lower side is the sum of ∠3 and ∠8 if they are adjacent, but they are at different points, so not directly.

Actually, the angle at T between the two transversals is the same as the angle between the directions, and it can be found from the angles at the top.

Perhaps the angle at T is equal to |m∠2 - m∠6| or something, but let's think differently.

Another standard way: the angle between two lines can be found from the angles they make with a third line.

For example, transversal c makes an angle with line a, say m∠1 or m∠2, and transversal d makes an angle with line a, say m∠5 or m∠6, then the angle between c and d is |angle_c - angle_d|.

But it's messy.

Let's use the given values directly.

Given m∠7 = 45°.

Suppose that m∠7 and m∠3 are corresponding angles or alternate.

Perhaps m∠7 and m∠11 are corresponding if lines are parallel.

Assume that.

If m∠7 = 45°, and if it corresponds to m∠11, then m∠11 = 45°.

Similarly, m∠14 = 85°, and if it corresponds to m∠6, then m∠6 = 85°.

Then at the top-right intersection, if m∠6 = 85°, and m∠7 = 45°, then they are adjacent, so 85 + 45 = 130 ≠ 180, so not adjacent, which is good, but then what is the relationship.

At the top-right intersection, the sum of angles around the point is 360°, and adjacent angles sum to 180°.

So if m∠6 = 85°, m∠7 = 45°, then if they are not adjacent, they could be vertical or something, but 85 and 45 are not equal, so not vertical.

Perhaps they are on the same side.

Let's calculate the other angles at that intersection.

Suppose at top-right intersection, the angles are:

Let’s say ∠5, ∠6, ∠7, ∠8 in order.

Then ∠5 + ∠6 = 180°, ∠6 + ∠7 = 180°, etc, only if they are adjacent.

In a circle, adjacent angles sum to 180° if on a straight line, but at a point, any two adjacent angles sum to 180° only if they form a linear pair, which they do if they are on a straight line.

At the intersection, each pair of adjacent angles forms a linear pair, so sum to 180°.

So for any two adjacent angles at the intersection, their sum is 180°.

So if m∠7 = 45°, then its adjacent angles are 135° each.

So if ∠6 is adjacent to ∠7, then m∠6 = 135°.

If ∠8 is adjacent to ∠7, then m∠8 = 135°.

Then the vertical angle to ∠7 is the one not adjacent, which is ∠5, so m∠5 = 45°.

Now, if m14 = 85°, and if it is at the bottom-right intersection, and if it is adjacent to ∠13, then m∠13 = 95°, etc.

Now, for parallel lines, corresponding angles are equal.

So, for example, ∠5 and ∠13 are corresponding if they are in the same relative position.

If 5 is at top, left of transversal d, above line a, then corresponding on bottom is ∠13, left of transversal d, above line b.

So if lines are parallel, m∠5 = m∠13.

But m∠5 = 45°, m∠13 = 95° (if m∠14 = 85° and adjacent), so 45 = 95, impossible.

Therefore, perhaps m∠14 is not adjacent to ∠13 in that way, or perhaps the correspondence is different.

Maybe m∠14 corresponds to ∠8 or ∠6.

Suppose that m∠14 corresponds to ∠6.

Then m∠6 = m∠14 = 85°.

But from m∠7 = 45°, and if 6 and ∠7 are adjacent, then m6 + m∠7 = 180°, so 85 + 45 = 130 ≠ 180, contradiction.

So ∠6 and ∠7 are not adjacent.

In that case, at the intersection, if ∠6 and ∠7 are not adjacent, they could be vertical, but 85 ≠ 45, so not.

Or they could be opposite in some way, but in a cross, only vertical angles are equal, and adjacent sum to 180.

So the only possibilities are that at each intersection, the angles are paired as (A,B) adjacent, sum 180, and (A,C) vertical, equal, etc.

So for two angles at the same intersection, if they are not vertical and not adjacent, it's impossible; they must be either adjacent or vertical.

In a plane, at an intersection of two lines, there are four angles: two pairs of vertical angles, and each angle is adjacent to two others.

So for any two distinct angles at the same intersection, they are either vertical (equal) or adjacent (sum 180°).

So for m∠6 and m∠7 at the same intersection, they must be either equal or sum to 180°.

Given m∠7 = 45°, if m∠6 = 85°, then 45 + 85 = 130 ≠ 180, and 45 ≠ 85, so impossible.

Therefore, m∠6 cannot be 85° if m7 = 45° at the same intersection.

So perhaps m∠14 does not correspond to an angle at the same intersection as m∠7.

Perhaps m∠7 and m∠14 are not related by correspondence directly.

Let's consider the triangle again.

In diagram A, the triangle has angles at:

- The left bottom intersection: the angle between line b and transversal c, which is inside the triangle. This is likely m∠10, because if the triangle is above the bottom line, then at left bottom, the angle above the line and between the transversal and the line is ∠10.

Similarly, at right bottom, the angle above the line and between the transversal and the line is m∠13.

At the intersection of the two transversals, the angle inside the triangle is the angle between them, say m∠X.

Then m∠10 + m13 + m∠X = 180°.

Now, m∠X is vertically opposite to the angle at the top between the two transversals on the lower side.

At the top, between the two transversals, on the lower side, that angle is the angle between transversal c and d, which can be found from the angles at the top.

In particular, at the top-left intersection, the angle on the lower-right side is m∠3.

At the top-right intersection, the angle on the lower-left side is m∠8.

Then, the angle between the two transversals at the top on the lower side is the sum of m∠3 and m∠8 if they are on the same side, but since the transversals are diverging, the angle between them is |m∠3 - m∠8| or something.

Actually, the angle between the two transversals is equal to the absolute difference of the angles they make with the horizontal, if the horizontal is the reference.

Assume that line a is horizontal.

Then, the angle that transversal c makes with line a is, say, m∠1 or m∠2.

Typically, the acute angle or the angle on a specific side.

For example, if at top-left intersection, m∠1 is the angle between line a and transversal c on the upper-left, then the angle on the lower-right is m∠3, and if the line is straight, m∠1 + m∠3 = 180° if they are adjacent, but they are vertical if numbered properly.

I think I need to guess the intended solution.

Let me search for a common pattern or assume values.

Suppose that m∠7 = 45° is the angle at the top, and it is equal to m∠3 by vertical or corresponding.

Perhaps m∠7 and m∠3 are vertical angles, but they are at different intersections.

Another idea: perhaps "m∠7" refers to the angle at position 7, and in the diagram, it is the angle that is alternate interior to m∠11 or something.

Let's calculate the answer based on standard problems.

I recall that in some similar problems, with m∠7 = 45°, m∠14 = 85°, then the triangle angles are 45°, 85°, and 50°, since 45+85+50=180.

So perhaps m∠10 = 45°, m∠13 = 85°, and the angle at T is 50°.

Then, the angle at T is 50°, so vertically opposite at the top is also 50°.

Then at the top, the angle between the two transversals on the lower side is 50°.

Then, at the top-left intersection, the angle on the lower-right side is m∠3, and at top-right, the angle on the lower-left side is m∠8, and m∠3 + m∠8 = 50° or |m∠3 - m∠8| = 50°, but likely they are on the same side, so sum.

Assume that m∠3 + m∠8 = 50°.

Also, at each intersection, angles sum appropriately.

For example, at top-left intersection, m∠1 + m∠2 = 180°, m∠2 + m∠3 = 180°, etc.

Suppose that m∠3 = x, m∠8 = y, x + y = 50°.

Then at top-left, m∠2 = 180 - x (since adjacent to ∠3)

At top-right, m∠6 = 180 - y (adjacent to ∠8)

Then, if lines are parallel, corresponding angles.

For example, m∠2 corresponds to m∠10, so m∠10 = m2 = 180 - x

But we assumed m∠10 = 45° for the triangle, so 180 - x = 45, so x = 135°, but then y = 50 - x = 50 - 135 = -85, impossible.

So not.

Perhaps m∠10 = m∠3 by alternate interior, if lines are parallel.

If lines a || b, then alternate interior angles are equal.

So for transversal c, m∠3 = m∠11 (alternate interior)

For transversal d, m∠8 = m∠14? Let's see.

If m∠8 is at top, lower-left, then alternate interior would be at bottom, upper-right, which is m∠14.

Yes! So if lines are parallel, then m∠8 = m∠14 = 85°.

Similarly, for transversal c, m∠3 = m∠11.

Now, from m∠7 = 45°, and at top-right intersection, m∠7 and m∠8 are adjacent, so m∠7 + m∠8 = 180°, so 45 + 85 = 130 ≠ 180, still contradiction.

Unless m∠7 and m∠8 are not adjacent.

In that case, if at top-right intersection, m∠7 and m∠8 are vertical, then m∠7 = m∠8, but 45 ≠ 85, so not.

So the only possibility is that m∠7 and m∠8 are not at the same intersection, but they are, according to standard numbering.

Perhaps the numbering is such that m∠7 is at a different location.

Let's look at the second diagram for clue, but the user only asked for this, and we have to solve both, but perhaps for diagram A, we can proceed with the triangle.

Assume that in the triangle, the angles are m∠9, m∠10, and the angle at T, but usually it's m∠10, m∠13, and the angle at T.

And given that m∠7 = 45° might be related to the angle at T.

Perhaps m∠7 is vertically opposite to the angle at T.

So if m∠7 = 45°, then the angle at T is 45°.

Then in the triangle, m∠10 + m∠13 + 45° = 180°, so m∠10 + m∠13 = 135°.

Also, m∠14 = 85°, and if m∠14 is at the right bottom, and if it is adjacent to m∠13, then m∠13 = 180 - 85 = 95°.

Then m∠10 = 135 - 95 = 40°.

Then, since lines are parallel, for transversal c, m∠3 = m∠11 (alternate interior)

At top-left, m∠3 and m∠2 are adjacent, etc.

Also, m∠10 = 40°, and if m∠10 corresponds to m2 or something.

For transversal c, m∠2 and m∠10 are corresponding if both on the same side.

If m2 is at top, upper-right, m∠10 at bottom, upper-right, then yes, corresponding, so m∠2 = m∠10 = 40°.

Then at top-left intersection, m∠2 + m∠3 = 180° (adjacent), so m∠3 = 180 - 40 = 140°.

Then m∠11 = m∠3 = 140° (alternate interior).

At top-left, m∠1 + m∠2 = 180°, so m∠1 = 180 - 40 = 140°.

m∠4 = m∠2 = 40° (vertical) or something.

Let's systematize.

From above:

- m∠7 = 45° (given)

- Assume that m∠7 is vertically opposite to the angle at T, so angle at T = 45°.

- m∠14 = 85° (given)

- Assume that at bottom-right intersection, m∠14 and m∠13 are adjacent, so m∠13 = 180 - 85 = 95°.

- In the triangle, angles are m∠10, m13, and angle at T = 45°, so m∠10 + 95 + 45 = 180, so m∠10 = 40°.

- Now, for parallel lines, transversal c: m∠2 and m∠10 are corresponding angles (both on the upper side, right of transversal), so m∠2 = m∠10 = 40°.

- At top-left intersection, m∠2 and m∠3 are adjacent, so m∠3 = 180 - 40 = 140°.

- Also, m∠1 and m∠2 are adjacent, so m1 = 180 - 40 = 140°.

- m∠4 and m∠2 are vertical, so m4 = m∠2 = 40°.

- For transversal c, alternate interior angles: m∠3 and m∠11 are alternate interior, so m∠11 = m∠3 = 140°.

- m∠9 and m∠11 are adjacent at bottom-left, so m∠9 = 180 - 140 = 40°.

- m∠12 and m10 are vertical or something; at bottom-left, m∠10 and m∠12 are vertical? Let's see.

At bottom-left intersection (b and c):

Angles: ∠9 (top-left), ∠10 (top-right), ∠11 (bottom-right), ∠12 (bottom-left)

So vertical angles: ∠9 and ∠11, 10 and ∠12.

So m∠12 = m∠10 = 40°.

m∠9 = m∠11 = 140°? No, vertical angles are equal, so if m∠9 and m∠11 are vertical, then m∠9 = m11, but we have m∠11 = 140°, m∠9 = 40°, contradiction.

Mistake.

If at bottom-left intersection, ∠9 and ∠11 are not vertical; typically, ∠9 and ∠11 are opposite if numbered sequentially.

If numbered clockwise: ∠9 (NW), ∠10 (NE), ∠11 (SE), ∠12 (SW)

Then vertical angles are ∠9 and ∠11, 10 and ∠12.

So m∠9 = m∠11, m10 = m∠12.

But from above, we have m∠10 = 40°, so m∠12 = 40°.

m∠11 = 140°, so m∠9 = 140°.

But earlier from the triangle, we have m∠10 = 40°, which is correct, and m∠9 is not in the triangle; the triangle angle at left bottom is m∠10, not m∠9.

In the triangle, the angle at left bottom is the angle between line b and transversal c, on the side towards the triangle, which is the upper side, so if the triangle is above the bottom line, then at left bottom, the angle above the line is between the line and the transversal, which is ∠10 if ∠10 is NE, and the triangle is to the right, so yes, m∠10 is the angle of the triangle at left bottom.

Similarly, at right bottom, the angle of the triangle is m∠13, which is NW or NE? If the triangle is to the left, then at right bottom, the angle above the line and to the left is m∠13 if it's NW.

In our case, we have m∠13 = 95°, m∠10 = 40°, angle at T = 45°, sum 40+95+45=180, good.

Now, at bottom-left intersection, m∠10 = 40°, and since vertical to m∠12, so m12 = 40°.

m∠9 and m∠11 are vertical, and m∠9 + m∠10 = 180° (adjacent), so m∠9 = 180 - 40 = 140°, so m∠11 = 140°.

Good.

Now for top-left intersection:

m∠1, m∠2, m∠3, m∠4

With m∠2 = 40° (corresponding to m∠10)

Then m∠1 + m∠2 = 180°, so m∠1 = 140°

m∠2 + m∠3 = 180°, so m∠3 = 140°

m∠3 + m∠4 = 180°, so m∠4 = 40°

Vertical angles: m∠1 and m∠3 are vertical? 140 and 140, yes, so m∠1 = m3 = 140°

m∠2 and m∠4 are vertical, 40 and 40, good.

Now for top-right intersection:

We have m∠7 = 45° (given)

Assume that m∠7 is at SE position.

Then its vertical angle is m∠5 = 45°

Adjacent angles: m∠6 and m∠8 are 135° each,