Supplementary and Congruent Angles online exercise for | Live ... - Free Printable

Let's solve the problem step by step.

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A) Find whether the following angles are supplementary or congruent.

We are given a diagram of two intersecting lines forming 8 angles labeled 1 through 8.

#### Key Definitions:
- Supplementary angles: Two angles that add up to 180°.
- Congruent angles: Angles that have the same measure (equal in size).

When two lines intersect:
- Vertical angles are congruent.
- Adjacent angles that form a straight line are supplementary.

Let’s analyze each pair:

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a) ∠1 and ∠2

- These are adjacent angles forming a straight line → they are supplementary.
- ✔ Supplementary

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b) ∠4 and ∠8

- ∠4 and ∠8 are vertical angles (opposite angles formed by intersection).
- Vertical angles are congruent.
- ✔ Congruent

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c) ∠2 and ∠7

- Let's trace: ∠2 is at the top-right, ∠7 is at the bottom-left.
- Are they vertical? No.
- Are they adjacent? No.
- But let's see their positions: ∠2 and ∠7 are not adjacent, nor are they opposite.
- Actually, ∠2 and ∠7 are not vertical or adjacent. But let's check if they are supplementary?

Wait — we can use the fact that:
- ∠2 and ∠3 are supplementary (they form a straight line).
- ∠3 and ∠6 are vertical → so ∠3 = ∠6
- ∠6 and ∠7 are supplementary (form a straight line)

But better way: Look at the diagram.

Actually, from standard angle relationships:

- ∠2 and ∠7: are they on a straight line? No.
- But ∠2 and ∠5 are vertical → equal.
- ∠5 and ∠7 are adjacent → together make a straight line? Yes! ∠5 and ∠7 form a straight line → supplementary.

So ∠2 = ∠5 (vertical), and ∠5 + ∠7 = 180° → so ∠2 + ∠7 = 180° → supplementary

✔ Supplementary

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d) ∠1 and ∠5

- ∠1 and ∠5 are vertical angles → opposite angles.
- So they are congruent.
- ✔ Congruent

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f) ∠3 and ∠6

- ∠3 and ∠6 are vertical angles → opposite angles.
- So they are congruent.
- ✔ Congruent

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g) ∠7 and ∠6

- These are adjacent angles forming a straight line → they are supplementary.
- ✔ Supplementary

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✔ Summary for Part A:

| Pair | Relationship |
|------|--------------|
| a) ∠1 and ∠2 | Supplementary |
| b) ∠4 and ∠8 | Congruent |
| c) ∠2 and ∠7 | Supplementary |
| d) ∠1 and ∠5 | Congruent |
| f) ∠3 and ∠6 | Congruent |
| g) ∠7 and ∠6 | Supplementary |

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B) Find the measure of the indicated angle.

Now we have a different diagram with two vertical lines and one diagonal line crossing them.

Angles are labeled 1 through 8.

Let’s interpret this carefully.

From the image description:
- There are two parallel vertical lines.
- A transversal (diagonal line) crosses both.
- This forms several angles.

We'll use:
- Vertical angles are congruent.
- Corresponding angles are congruent (if lines are parallel).
- Supplementary angles add to 180°.
- Linear pairs are supplementary.

Let’s go through each part.

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#### a) If m∠3 = 143°, find the measure of ∠4.

- ∠3 and ∠4 are adjacent and form a straight line → linear pair.
- So:
$$
m∠3 + m∠4 = 180^\circ \\
143^\circ + m∠4 = 180^\circ \\
m∠4 = 180^\circ - 143^\circ = 37^\circ
$$

✔ m∠4 = 37°

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#### b) If m∠1 = 143°, find the measure of ∠7.

- First, identify ∠1 and ∠7.
- ∠1 is at the bottom right (near the lower vertical line).
- ∠7 is at the upper left (on the other vertical line).
- Since the vertical lines are parallel and cut by a transversal, we look for corresponding angles.

Let’s see:
- ∠1 and ∠5 are vertical angles → ∠5 = ∠1 = 143°
- Now, ∠5 and ∠7: are they corresponding?
- ∠5 is above the transversal, on the left side.
- ∠7 is also above the transversal, on the left side.
- Wait: ∠7 is below the transversal?

Wait — let's recheck based on standard labeling.

Looking at the diagram (as described):
- The transversal crosses the two vertical lines.
- At the lower intersection, angles 1, 2, 3, 4 are formed.
- At the upper intersection, angles 5, 6, 7, 8 are formed.

Assuming:
- ∠1 and ∠5 are corresponding angles (both are on the same side of the transversal and same relative position to the vertical lines).

But ∠1 is at the bottom right, ∠5 is at the top left → no, that’s not correct.

Wait: likely:
- ∠1 and ∠5 are vertical angles? No, they’re on different intersections.

Actually, in such diagrams:
- ∠1 and ∠5 are corresponding angles if the lines are parallel.

But wait — ∠1 is at the bottom right, ∠5 is at the top left — not matching.

Let’s think differently.

Standard convention:
- When a transversal crosses two parallel lines:
- Corresponding angles: e.g., ∠1 and ∠5 (if ∠1 is bottom right, ∠5 is top right) — but here ∠5 is labeled on the upper left.

Wait — from the diagram:
- At the bottom intersection:
- ∠1: bottom right
- ∠2: bottom left
- ∠3: top left
- ∠4: top right

At the top intersection:
- ∠5: top right
- ∠6: top left
- ∠7: bottom left
- ∠8: bottom right

Wait — actually, from the labels:
- ∠1 is at bottom right (of lower intersection)
- ∠2 is at bottom left
- ∠3 is at top left
- ∠4 is at top right

Then at the upper intersection:
- ∠5 is at top right
- ∠6 is at top left
- ∠7 is at bottom left
- ∠8 is at bottom right

So:
- ∠1 and ∠5 are not on the same side.

But notice:
- ∠1 and ∠5 are both on the right side of the transversal, and below and above respectively → corresponding angles.

Yes! So:
- ∠1 and ∠5 are corresponding angles → since lines are parallel, they are congruent.

But wait — ∠1 is at bottom right, ∠5 is at top right → yes, same side, same relative position → corresponding.

So:
- m∠1 = 143° → m∠5 = 143°

Now, ∠7 is at the bottom left of the upper intersection.

We need m∠7.

But ∠7 and ∠5 are adjacent and form a straight line? No — they are on the same side of the transversal.

Wait: at the upper intersection:
- ∠5 (top right) and ∠7 (bottom left) are not adjacent.

Actually, ∠5 and ∠7 are vertical angles? No.

Wait — at the upper intersection:
- ∠5 and ∠7 are opposite angles? No.

Let’s label properly:

At the upper intersection:
- The transversal cuts the top vertical line.
- Angles:
- Top right: ∠5
- Top left: ∠6
- Bottom left: ∠7
- Bottom right: ∠8

So:
- ∠5 and ∠7 are not adjacent; they are opposite across the vertex? No — ∠5 is top right, ∠7 is bottom left → they are vertical angles → yes!

So ∠5 and ∠7 are vertical angles → congruent.

So:
- m∠5 = 143° → m∠7 = 143°

✔ m∠7 = 143°

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#### c) If m∠2 = 37°, find the measure of ∠6.

First, where is ∠2?
- ∠2 is at the bottom left of the lower intersection.

Now, ∠6 is at the top left of the upper intersection.

Are ∠2 and ∠6 related?

Look at their positions:
- Both are on the left side of the transversal.
- ∠2 is below, ∠6 is above → they are corresponding angles.

Since the lines are parallel, corresponding angles are congruent.

So:
- m∠2 = 37° → m∠6 = 37°

✔ m∠6 = 37°

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#### d) If m∠1 = 143°, find the measure of ∠5.

Earlier we said:
- ∠1 and ∠5 are corresponding angles → congruent.

So:
- m∠1 = 143° → m∠5 = 143°

✔ m∠5 = 143°

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✔ Final Answers:

#### Part A:
a) ∠1 and ∠2 are supplementary
b) ∠4 and ∠8 are congruent
c) ∠2 and ∠7 are supplementary
d) ∠1 and ∠5 are congruent
f) ∠3 and ∠6 are congruent
g) ∠7 and ∠6 are supplementary

#### Part B:
a) m∠4 = 37°
b) m∠7 = 143°
c) m∠6 = 37°
d) m∠5 = 143°

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Let me know if you'd like a visual explanation or diagram labeling!