Geometry worksheet with angle calculations for quadrilaterals, showing diagrams and solved examples.

Let’s solve each problem step by step. We’re dealing with kites — quadrilaterals with two pairs of adjacent equal sides. In a kite:

- One diagonal is a line of symmetry (the vertical one in these diagrams).
- The angles between unequal sides are equal.
- The diagonals intersect at right angles (90°).
- The sum of all interior angles in any quadrilateral is 360°.

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Problem 4:

We have kite DEFG, with diagonals intersecting at H.

Given: ∠D = 59° (that’s ∠EDG or ∠FDG? Looking at diagram, it’s angle at D, so ∠EDF or ∠GDF? Actually, label says “59°” near vertex D, and arrow points to angle between side DG and diagonal DH — so that’s ∠GDH = 59°.

But wait — looking again: the 59° is labeled at vertex D, and since it’s a kite, and diagonal DF is horizontal, EG is vertical, intersecting at H.

Actually, standard notation: in kite DEFG, vertices are D, E, F, G. Diagonals are DF and EG, crossing at H.

Angle at D is split by diagonal DF? No — actually, in this diagram, diagonal EG is the axis of symmetry? Wait — let’s look carefully.

In problem 4, the kite has vertices D, E, F, G. Diagonal EG is drawn vertically, DF horizontally, intersecting at H.

The 59° is marked at vertex D, between side DG and diagonal DH — so that’s ∠GDH = 59°.

Since diagonals of a kite intersect at 90°, then ∠DHG = 90°, ∠DHE = 90°, etc.

Also, because EG is the axis of symmetry (assuming it’s the main diagonal), then triangle DGH ≅ triangle FGH, and triangle DEH ≅ triangle FEH.

Wait — actually, in a kite, only one diagonal is the axis of symmetry — usually the one connecting the vertices where the equal sides meet. Here, if DE = DG and FE = FG, then diagonal DF would be the axis? But in the diagram, EG is drawn as the vertical diagonal, and it looks like it’s the symmetry axis.

Looking at the labels: angle at D is 59°, and it’s on the left side. Since it’s a kite, and assuming symmetry over EG, then angle at F should also be 59°? But no — in a kite, the angles between the unequal sides are equal. So if DE = DG and FE = FG, then angles at E and G are the ones that may differ, and angles at D and F are the ones that are equal? Actually, no — standard property: in a kite, one pair of opposite angles are equal — specifically, the angles between the unequal sides.

Actually, let me recall: in kite ABCD with AB=AD and CB=CD, then angles at B and D are not necessarily equal; rather, angles at A and C — wait, no.

Better approach: use triangle properties.

In problem 4:

Diagonals intersect at H at 90°.

At vertex D, we’re told ∠GDH = 59°. That’s the angle between side DG and diagonal DH.

Since diagonal EG is perpendicular to DF, then in triangle DGH, we have:

∠GDH = 59°, ∠DHG = 90°, so ∠DGH = 180° - 59° - 90° = 31°.

Similarly, since the kite is symmetric over diagonal EG (assuming that’s the axis), then triangle DGH ≅ triangle FGH, so ∠GFH = 59°, ∠FGH = 31°.

Now, what about the top part? Triangle DEH and FEH.

We don’t have direct info, but perhaps we can find other angles.

The question asks for:

m∠GDE = ?
m∠DEH = ?
m∠DGH = ?

First, m∠DGH: from above, in triangle DGH, angles are 59° at D, 90° at H, so 31° at G. So ∠DGH = 31°.

But ∠DGH is the same as ∠DGE? Yes, since H is on EG.

So m∠DGH = 31°.

Now, m∠GDE: that’s the angle at D between points G, D, E. So that’s the full angle at vertex D.

In the kite, at vertex D, the angle is composed of ∠GDH and ∠EDH.

We know ∠GDH = 59°. What is ∠EDH?

Since the kite is symmetric over EG, and assuming DE = DG, then triangles DEH and DGH are congruent? Only if EH = GH, which may not be true.

Actually, in a kite, the diagonal that is the axis of symmetry bisects the angles at its endpoints.

If EG is the axis of symmetry, then it bisects angles at E and G.

But at D and F, it does not necessarily bisect.

However, in this case, since we have ∠GDH = 59°, and if the kite is symmetric over EG, then ∠EDH should also be 59°, because D is on the axis? No, D is not on the axis; EG is the axis, so D and F are symmetric.

Actually, if EG is the axis of symmetry, then point D reflects to point F, so angle at D should equal angle at F.

And diagonal EG bisects the angles at E and G.

But for angle at D, it is not bisected by EG unless specified.

In this diagram, the 59° is given as the angle between DG and DH, and since DH is part of diagonal DF, and EG is perpendicular, then in triangle DGH, we have angles 59°, 90°, 31° as before.

Now, for triangle DEH: we don't know any angles yet.

But perhaps we can find the full angle at D.

Notice that the full angle at D, ∠GDE, is the angle between sides DG and DE.

Since diagonal DF is drawn, and it splits angle D into two parts: ∠GDF and ∠EDF.

From the diagram, ∠GDF = 59°, and if the kite is symmetric, then ∠EDF should also be 59°, so total angle at D is 118°.

Is that correct? Let me verify.

In a kite with DE = DG and FE = FG, then diagonal DF is the axis of symmetry? Or EG?

Standard definition: a kite has two pairs of adjacent equal sides. So possible configurations:

- DE = DG and FE = FG → then diagonal DF is the axis of symmetry, and it bisects angles at D and F.

- Or DE = EF and DG = GF → then diagonal EG is the axis.

In this diagram, since EG is drawn as the vertical diagonal and it's likely the symmetry axis, and D and F are symmetric, so probably DE = FE and DG = FG, meaning that E and G are the "top" and "bottom", and D and F are the "sides".

In that case, diagonal EG is the axis of symmetry, so it bisects angles at E and G, and also, angles at D and F are equal.

Moreover, diagonal EG is perpendicular to DF, and bisects DF? Not necessarily, but in this case, since it's symmetric, yes, H is midpoint of DF.

But for angles: at vertex D, the angle is between sides DE and DG.

Diagonal DF is drawn, but it doesn't necessarily bisect angle D.

However, in the diagram, the 59° is labeled as the angle between DG and DH, and DH is along DF.

Since the kite is symmetric over EG, and D and F are symmetric, then the angle between DG and DF should equal the angle between FE and FD, but not directly helpful.

Perhaps I should consider triangle DGH and triangle DEH.

We know in triangle DGH: ∠GDH = 59°, ∠DHG = 90°, so ∠DGH = 31°.

Now, for triangle DEH: we don't know any angles, but if we assume that the kite is convex and symmetric, then perhaps ∠EDH = ∠GDH = 59°, because of symmetry? But that would mean that diagonal DF bisects angle D, which is true only if DE = DG, but in this configuration, if EG is the axis, then DE = FE and DG = FG, so DE may not equal DG.

This is confusing. Let me look back at the diagram description.

In problem 4, the kite is labeled D, E, F, G, with E at top, G at bottom, D left, F right. Diagonals EG and DF intersect at H. Angle at D is marked as 59°, and it's the angle between side DG and diagonal DH. So that's ∠GDH = 59°.

Since diagonals are perpendicular, ∠DHG = 90°.

So in triangle DGH, angles sum to 180°, so ∠DGH = 180 - 59 - 90 = 31°.

Now, m∠DGH is asked, which is this 31°.

Next, m∠GDE: this is the angle at D between G, D, E, so that's the full angle at vertex D, which is ∠GDE = ∠GDH + ∠HDE.

What is ∠HDE? In triangle DEH, we have points D, E, H.

We know that ∠DHE = 90°, since diagonals are perpendicular.

But we don't know other angles yet.

However, because the kite is symmetric over EG, and D and F are symmetric, then triangle DEH should be congruent to triangle FEH, and triangle DGH congruent to triangle FGH.

Moreover, since DG = FG (because symmetric over EG), and DE = FE, then yes.

Now, at vertex D, the angle ∠GDE is composed of ∠GDH and ∠EDH.

By symmetry, since the figure is symmetric over EG, and H is on EG, then the angle between DG and EG should equal the angle between FG and EG, but for D, it's different.

Actually, the ray DH is along DF, and since DF is perpendicular to EG, and by symmetry, the angle that DG makes with DF should be the same as the angle that DE makes with DF? No.

Let's think: from point D, we have rays to G, to E, and to H (which is on DF).

The angle between DG and DH is 59°.

The angle between DE and DH is unknown.

But in the kite, the sides are DG and DE, and they are not necessarily equal.

However, in many kites, especially in such problems, often the diagonal from the "vertex" bisects the angle, but here it's not specified.

Perhaps I can use the fact that the sum of angles around point D is 360°, but that's not helpful.

Another idea: perhaps the 59° is the entire angle at D? But the diagram shows it as part of it.

Looking back at the user's image description, in problem 4, it says "59°" at D, and it's likely the angle of the triangle, but let's read the question: it asks for m∠GDE, which is the angle at D in the kite, so probably the full angle.

But in the diagram, it's marked as 59° for the smaller angle.

Perhaps in this context, since no other information, and by symmetry, we can assume that diagonal DF bisects angle D, so ∠GDE = 2 * 59° = 118°.

That makes sense, and is common in such problems.

Moreover, in the answer choices or expected answers, it might be that.

Let me check with the other parts.

We also need m∠DEH.

In triangle DEH, if we can find angles.

Assume that by symmetry, since the kite is symmetric over EG, then angle at D equals angle at F, and angle at E equals angle at G? No, in a kite, only one pair of opposite angles are equal — the ones between the unequal sides.

In this case, if DE = FE and DG = FG, then the equal angles are at D and F.

So ∠D = ∠F.

And the diagonal EG bisects angles at E and G.

So for angle at D, it is not bisected by EG, but by DF? Not necessarily.

However, in this specific diagram, since the 59° is given as ∠GDH, and if we assume that DF is the angle bisector, then ∠GDE = 2*59° = 118°.

Otherwise, we don't have enough information.

Perhaps from the context of the worksheet, it's intended that way.

Let's look at problem 6 for clue, but let's try to proceed.

Another thought: in triangle DGH, we have angles 59°, 90°, 31°.

Now, for triangle DEH, if we knew something, but we don't.

But perhaps the full angle at G is needed.

m∠DGH is already found as 31°, and that's part of angle at G.

Angle at G is ∠DGF, which is composed of ∠DGH and ∠FGH.

By symmetry, since triangle DGH ≅ triangle FGH, then ∠FGH = ∠DGH = 31°, so full angle at G is 62°.

Similarly, at D, if we had symmetry, but for angle at D, it's between DG and DE, and if we assume that the diagonal DF bisects it, then 118°.

Perhaps the 59° is meant to be the angle of the triangle, and we need to find the rest.

Let's calculate m∠DEH.

In triangle DEH, we have points D, E, H.

We know that ∠DHE = 90°.

If we can find another angle.

Notice that at vertex E, the angle is split by diagonal EG into two parts: ∠DEH and ∠FEH.

By symmetry, since the kite is symmetric over EG, then ∠DEH = ∠FEH, and also, the diagonal EG bisects angle at E, so ∠DEG = ∠FEG, but since H is on EG, then ∠DEH = ∠FEH.

But we don't know the value.

Perhaps we can use the fact that the sum of angles in the kite is 360°.

Let me denote:

Let ∠D = ∠F = x (since they are the equal angles in the kite).

Let ∠E = y, ∠G = z.

Then x + x + y + z = 360°, so 2x + y + z = 360°.

Also, since EG is the axis of symmetry, it bisects angles at E and G, so ∠DEH = y/2, ∠DGH = z/2.

From earlier, in triangle DGH, we have ∠GDH = 59°, which is part of angle at D.

Angle at D is x, and it is composed of ∠GDH and ∠EDH.

If we assume that diagonal DF bisects angle D, then ∠GDH = ∠EDH = x/2 = 59°, so x = 118°.

Then from 2x + y + z = 360°, 2*118 + y + z = 360, so 236 + y + z = 360, so y + z = 124°.

Also, in triangle DGH, we have angles: at D: 59°, at H: 90°, at G: 31°, and this 31° is half of angle at G, since EG bisects angle G, so z/2 = 31°, so z = 62°.

Then y + 62 = 124, so y = 62°.

Then angle at E is 62°, so ∠DEH = y/2 = 31°.

And m∠GDE = x = 118°.

m∠DGH = 31°, as before.

So everything consistent.

So for problem 4:

m∠GDE = 118°
m∠DEH = 31°
m∠DGH = 31°

Now, let's confirm with the calculations.

In triangle DGH: ∠GDH=59°, ∠DHG=90°, ∠DGH=31°.

Since EG bisects angle G, full angle at G is 62°.

By symmetry, angle at F is same as at D, 118°.

Sum: D+F+E+G = 118+118+62+62=360°, good.

In triangle DEH: ∠DHE=90°, ∠DEH=31° (since half of 62°), so ∠EDH = 180-90-31=59°, which matches the assumption that DF bisects angle D, so ∠EDH=59°, same as ∠GDH.

Perfect.

So answers for problem 4:

m∠GDE = 118°
m∠DEH = 31°
m∠DGH = 31°

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Problem 6:

Kite with vertices, let's say top, bottom, left, right. Diagonals intersect at center.

Labels: angles marked: at left vertex, 73° and angle 1; at right vertex, 46° and angle 5; at top, angles 6 and 7; at bottom, angles 2 and 3; and at intersection, angles 4 and others, but angle 4 is at the intersection on the right side.

Specifically:

- At left vertex: angle between left side and horizontal diagonal is 73°, and angle between left side and vertical diagonal is angle 1.

- At right vertex: angle between right side and horizontal diagonal is 46°, and angle between right side and vertical diagonal is angle 5.

- At top vertex: angle between top-left side and vertical diagonal is angle 6, between top-right side and vertical diagonal is angle 7.

- At bottom vertex: angle between bottom-left side and vertical diagonal is angle 2, between bottom-right side and vertical diagonal is angle 3.

- At intersection point: angle between horizontal and vertical diagonals is 90°, and angle 4 is probably the angle in the bottom-right triangle or something. Looking at the diagram description, angle 4 is likely at the intersection, in the bottom-right quadrant.

Actually, in the text: "73°" at left, "46°" at right, and numbers 1,2,3,4,5,6,7 for other angles.

Typically, in such diagrams, the diagonals divide the kite into four triangles.

Let me denote the intersection point as O.

So, triangles: top-left, top-right, bottom-left, bottom-right.

At left vertex: the angle is split into two parts: one is 73° (between left side and horizontal diagonal), and the other is angle 1 (between left side and vertical diagonal).

Similarly at right: 46° between right side and horizontal diagonal, angle 5 between right side and vertical diagonal.

At top: angle 6 between top-left side and vertical diagonal, angle 7 between top-right side and vertical diagonal.

At bottom: angle 2 between bottom-left side and vertical diagonal, angle 3 between bottom-right side and vertical diagonal.

At intersection O: the angles are all 90° since diagonals are perpendicular, but angle 4 is probably labeled in one of the triangles, perhaps in the bottom-right triangle, the angle at O.

In the list, it asks for m∠1 to m∠6, and also m∠4, m∠5, etc.

Specifically: m∠1, m∠2, m∠3, m∠4, m∠5, m∠6.

Note that angle 7 is not asked, but might be needed.

First, in the bottom-left triangle: vertices: left vertex, bottom vertex, intersection O.

At left vertex: angle is angle 1 (between left side and vertical diagonal).

At bottom vertex: angle is angle 2 (between bottom-left side and vertical diagonal).

At O: angle is 90°.

Sum of angles in triangle is 180°, so angle 1 + angle 2 + 90° = 180°, so angle 1 + angle 2 = 90°.

Similarly, in bottom-right triangle: at right vertex: angle 5, at bottom vertex: angle 3, at O: 90°, so angle 5 + angle 3 = 90°.

In top-left triangle: at left vertex: 73°, at top vertex: angle 6, at O: 90°, so 73° + angle 6 + 90° = 180°, so angle 6 = 180 - 90 - 73 = 17°.

Similarly, in top-right triangle: at right vertex: 46°, at top vertex: angle 7, at O: 90°, so 46° + angle 7 + 90° = 180°, so angle 7 = 180 - 90 - 46 = 44°.

Now, at the left vertex, the full angle is composed of the angle in top-left triangle and bottom-left triangle, so 73° + angle 1.

Similarly, at right vertex: 46° + angle 5.

At top vertex: angle 6 + angle 7 = 17° + 44° = 61°.

At bottom vertex: angle 2 + angle 3.

In a kite, typically, the angles between the unequal sides are equal. Assuming that the kite has symmetry over the vertical diagonal, then left and right are symmetric, so angle at left should equal angle at right, and angle at top equals angle at bottom? Not necessarily.

In this case, since the given angles at left and right are different (73° and 46°), probably the symmetry is over the horizontal diagonal? But usually vertical.

Perhaps it's symmetric over the vertical diagonal, so left and right should be mirror images, but 73° ≠ 46°, contradiction.

Unless the 73° and 46° are not corresponding.

Perhaps the kite is not symmetric in that way.

Another possibility: in a kite, two pairs of adjacent sides equal. Suppose the top two sides are equal, and bottom two sides are equal, then the vertical diagonal is the axis of symmetry, so left and right should be symmetric, so the angles should be equal, but 73° ≠ 46°, so that can't be.

Perhaps the left and right sides are equal, and top and bottom are different, but then symmetry over horizontal diagonal.

Assume symmetry over the horizontal diagonal DF.

Then, top and bottom are symmetric, so angle at top equals angle at bottom, and angle at left equals angle at right? No.

If symmetric over horizontal diagonal, then reflecting over horizontal axis swaps top and bottom, so angle at top equals angle at bottom, and the left and right are mapped to themselves, so angles at left and right may not be equal.

In this case, at left vertex, the angle is split into 73° (with horizontal) and angle 1 (with vertical).

After reflection over horizontal axis, the 73° part would map to the bottom part, but at left vertex, the bottom part is angle 1, so if symmetric, then 73° should equal angle 1? But that might not be.

Let's think.

If the kite is symmetric over the horizontal diagonal, then the figure above the horizontal diagonal is mirror image of below.

So, for example, the top-left triangle is mirror image of bottom-left triangle.

Therefore, in top-left triangle, angle at left vertex is 73°, so in bottom-left triangle, angle at left vertex should be the same, 73°, but in bottom-left triangle, the angle at left vertex is angle 1, so angle 1 = 73°.

Similarly, in top-right triangle, angle at right vertex is 46°, so in bottom-right triangle, angle at right vertex should be 46°, which is angle 5, so angle 5 = 46°.

Then, in bottom-left triangle: angles: at left: angle 1 = 73°, at O: 90°, so at bottom: angle 2 = 180 - 73 - 90 = 17°.

Similarly, in bottom-right triangle: at right: angle 5 = 46°, at O: 90°, so at bottom: angle 3 = 180 - 46 - 90 = 44°.

Now, at bottom vertex, full angle is angle 2 + angle 3 = 17° + 44° = 61°.

At top vertex, from earlier, angle 6 + angle 7 = 17° + 44° = 61°, same as bottom, good.

At left vertex, full angle is 73° + angle 1 = 73° + 73° = 146°.

At right vertex, 46° + angle 5 = 46° + 46° = 92°.

Sum: 146 + 92 + 61 + 61 = let's calculate: 146+92=238, 61+61=122, total 238+122=360°, perfect.

Now, what about angle 4? In the diagram, angle 4 is probably at the intersection in the bottom-right triangle or something. Looking at the labels, it might be the angle at O in one of the triangles, but since all are 90°, perhaps it's labeled differently.

In the problem, it asks for m∠4, and in the list, it's included.

In some diagrams, angle 4 might be the angle in the bottom-right triangle at the bottom vertex or something, but we have angle 3 there.

Perhaps angle 4 is at the intersection, but it's 90°.

Looking back at the user's input: in problem 6, it says "angle 4" and it's likely the angle at the intersection point in the bottom-right quadrant, but since diagonals are perpendicular, it should be 90°.

But in the answer, probably it's 90°.

Perhaps it's labeled as the angle between the diagonals, which is always 90°.

In the list, it asks for m∠4, and from context, likely it's 90°.

To confirm, in the bottom-right triangle, angles are: at right vertex: 46°, at bottom vertex: angle 3=44°, at O: 90°, so if angle 4 is at O, it's 90°.

Similarly, in other places.

So, summarizing for problem 6:

m∠1 = 73° (from symmetry, equals the 73° at top-left)

m∠2 = 17° (in bottom-left triangle)

m∠3 = 44° (in bottom-right triangle)

m∠4 = 90° (at intersection, assuming it's the angle between diagonals)

m∠5 = 46° (from symmetry, equals the 46° at top-right)

m∠6 = 17° (calculated earlier from top-left triangle)

We also have m∠7=44°, but not asked.

So answers:

m∠1 = 73°
m∠2 = 17°
m∠3 = 44°
m∠4 = 90°
m∠5 = 46°
m∠6 = 17°

Now, to box the final answers.

But the user has multiple problems, but in the response, we need to provide answers for the ones that are incomplete or asked.

In the original request, it seems we need to solve the problems shown, particularly problem 4 and 6, as others have answers filled.

But to be complete, let's list all required.

For problem 4:

m∠GDE = 118°
m∠DEH = 31°
m∠DGH = 31°

For problem 6:

m∠1 = 73°
m∠2 = 17°
m∠3 = 44°
m∠4 = 90°
m∠5 = 46°
m∠6 = 17°

Now, Final Answer section should contain these.

But since it's multiple, perhaps list them clearly.

Since the instruction is to provide final answer after reasoning, and for each problem.

But in the format, it says "Final Answer: [provide only the final correct answer]", so probably for each subproblem.

To make it clear, I'll write:

For Problem 4:
m∠GDE = 118°
m∠DEH = 31°
m∠DGH = 31°

For Problem 6:
m∠1 = 73°
m∠2 = 17°
m∠3 = 44°
m∠4 = 90°
m∠5 = 46°
m∠6 = 17°

But perhaps just the values.

Since the user might expect the answers in order.

I think it's fine.

Now, to present.