Student studying kite properties in a math worksheet.

Let’s solve Problem 1 step by step.

We are given a kite with angles labeled:

- Angle at A = 60°
- Angle at B = ? (we’ll call it x)
- Angle at C = 64°
- Angle at D = ? (we’ll call it y)

But wait — looking at the diagram, it seems like the kite is labeled ABCD, and we’re told:

> m∠A = 60°, m∠C = 64°, and we need to find m∠B and m∠D.

Also, from the properties of a kite (written above):

→ One pair of opposite angles are congruent. Specifically, ∠ABC ≅ ∠ADC? Wait — no, let’s read carefully.

Actually, in the notes section, it says:

> “One pair of opposite angles are congruent: (∠ABC ≅ ∠ADC)”

Wait — that can’t be right for all kites. Actually, in a kite, only one pair of opposite angles are congruent — and they are the ones between the unequal sides. But more accurately:

In a kite, the angles between the *equal* sides are equal? No — let me recall properly.

Actually, standard property: In a kite, one pair of opposite angles are congruent — specifically, the angles that are between the two pairs of adjacent equal sides. More precisely:

If AB = AD and CB = CD (so A and C are the vertices where the equal sides meet), then angles at B and D are NOT necessarily equal — actually, angles at A and C are not necessarily equal either.

Wait — correction: In a kite, the angles between the *unequal* sides are equal? Let me think differently.

Better approach: Use the fact that the sum of interior angles of any quadrilateral is 360°.

So:

m∠A + m∠B + m∠C + m∠D = 360°

Given:

m∠A = 60°
m∠C = 64°
So: 60 + m∠B + 64 + m∠D = 360
→ m∠B + m∠D = 360 - 60 - 64 = 236°

Now, what else do we know about kites?

From the worksheet notes:

> “One pair of opposite angles are congruent: (∠ABC ≅ ∠ADC)”

Wait — ∠ABC is angle at B, ∠ADC is angle at D. So if those are congruent, then m∠B = m∠D.

Is that correct? Let me verify.

Actually, in a typical kite shaped like a diamond with top vertex A, bottom vertex C, left B, right D, and if AB=AD and CB=CD, then the axis of symmetry is AC. Then angles at B and D should be equal because of symmetry.

Yes! That makes sense. So in this case, since it's symmetric across diagonal AC, angles at B and D are equal.

Therefore:

m∠B = m∠D

And we have:

m∠B + m∠D = 236°
→ 2 × m∠B = 236°
→ m∠B = 118°
→ m∠D = 118°

Wait — but in the student’s work, they wrote:

“58 / 2 = 29” — that doesn’t make sense here.

Then they wrote “m∠B = 116°”, “m∠D = 116°”

How did they get 116?

Let me check their calculation:

They wrote:

Sum = 360
Then: 60 + 64 = 124
360 - 124 = 236 → that part is correct.

Then they wrote: “58 / 2 = 29” — where did 58 come from?

Wait — maybe they misread the diagram?

Looking again at the diagram in the image (even though I’m not supposed to describe it, I must interpret the numbers correctly).

The user has written on the paper:

For problem 1:

Angle at top (A) = 60°
Angle at bottom (C) = 64°
Left angle (B) = ?
Right angle (D) = ?

And they computed:

60 + 64 = 124
360 - 124 = 236
Then they wrote “58 / 2 = 29” — that seems wrong.

Wait — perhaps they thought the difference or something else?

Alternatively, maybe they confused which angles are equal.

Another possibility: In some kites, the angles at the ends of the symmetry axis are equal? No.

Standard rule: In a kite, exactly one pair of opposite angles are equal — and they are the ones that are between the two different-length sides. Actually, let’s clarify with a reliable source in mind.

Upon recalling: In a kite, the angles that are between the *congruent sides* are equal? No.

Actually, here’s the correct property:

In a kite, the angles formed by the two pairs of adjacent congruent sides are equal? Not quite.

Better: The kite has an axis of symmetry along one diagonal. The two angles that lie on the axis of symmetry are not necessarily equal, but the other two angles (the ones off the axis) are equal.

In standard labeling: If diagonal AC is the axis of symmetry, then angles at B and D are equal.

Yes — so in this case, angles at B and D are equal.

So m∠B = m∠D

Thus:

60 + 64 + x + x = 360
124 + 2x = 360
2x = 236
x = 118

So each of angles B and D is 118 degrees.

But why did the student write 116?

Perhaps they made a subtraction error.

360 - 60 - 64:

360 - 60 = 300
300 - 64 = 236 — correct.

236 ÷ 2 = 118 — not 116.

Unless... did they use different values?

Wait — looking back at the student’s writing:

They wrote:

“m∠A = 60°”
“m∠C = 64°”
Then below: “58 / 2 = 29” — that 58 might be a mistake.

Then they have “m∠B = 116°”, “m∠D = 116°”

How to get 116?

If they did 360 - 60 - 64 = 236, then 236 / 2 = 118 — not 116.

Unless they used 62 instead of 64? 360 - 60 - 62 = 238, 238/2=119 — no.

Or 360 - 60 - 68 = 232, 232/2=116 — ah! Maybe they misread angle C as 68° instead of 64°?

But in the diagram, it clearly says 64°.

Perhaps there’s another interpretation.

Another thought: Maybe the 60° and 64° are not both vertex angles? Or perhaps one is half-angle?

No, the diagram shows full angles.

Wait — in the student’s work, they also wrote “232°” next to the kite — that might be the sum of two angles?

Let me see: They have “232°” written near the bottom, and then “m∠B = 116°”, “m∠D = 116°”

232 / 2 = 116 — so they must have calculated the sum of angles B and D as 232°.

How? 360 - 60 - 68 = 232 — so if they took angle C as 68°, but it’s written as 64°.

Perhaps it’s a typo in my reading? Or in the original?

Since the problem is presented as is, and assuming the diagram shows 60° and 64°, then the correct answer should be 118° each.

But let’s double-check the kite property.

I recall now: In a kite, the angles between the unequal sides are equal? No.

Standard definition: A kite has two pairs of adjacent congruent sides. The angles between the congruent sides are called the "vertex angles", and the other two are "non-vertex angles". And the non-vertex angles are equal.

More precisely: The angles that are between the two different-length sides are equal? Let's think.

Example: Suppose AB = AD, and CB = CD. Then triangle ABC and ADC are not necessarily congruent, but triangles ABD and CBD may have symmetries.

Actually, the key is: Diagonal AC is the perpendicular bisector of BD, and also bisects angles at A and C.

But for angles: Angles at B and D are equal only if the kite is convex and symmetric, which it is.

To confirm, let's calculate with actual values.

Suppose angle A = 60°, angle C = 64°, and angles B and D are equal.

Sum: 60 + 64 + x + x = 360 → 124 + 2x = 360 → 2x = 236 → x = 118.

If the student got 116, they likely subtracted incorrectly: 360 - 60 = 300, 300 - 64 = 236 — that's correct. 236 divided by 2 is 118.

Perhaps they did 360 - 60 - 64 = 236, then 236 - 4 = 232 for some reason, then /2 = 116.

Or maybe they added 60 + 64 = 124, then 360 - 124 = 236, but then thought 236 is for three angles or something.

Another idea: Perhaps the 60° and 64° are not the full angles, but half-angles? But the diagram doesn't suggest that.

Looking at the student's work: They have "58 / 2 = 29" — where does 58 come from? 60 - 2? 64 - 6? Not clear.

Perhaps they found the difference: 64 - 60 = 4, then did something with that.

I think the most reasonable conclusion is that the student made an arithmetic error, and the correct answer is 118° for both angles B and D.

But let's consider: Is it possible that in this kite, the equal angles are at A and C? No, because 60 ≠ 64, so they can't be equal.

The property says "one pair of opposite angles are congruent", so since A and C are not equal, it must be B and D that are equal.

Therefore, m∠B = m∠D = 118°.

Final Answer:
m∠B = 118°, m∠D = 118°