Question

find $m\angle 6$.

Explanation:

Step1: Analyze right triangle for ∠1

In the right triangle with 25° and right angle, ∠1 = 90° - 25° = 65°? Wait, no, wait. Wait, the triangle with 51° and right angle? Wait, no, let's look at the angles. Wait, the angle adjacent to 51°: in the triangle with 51° and right angle, ∠3 + 51° + 90° = 180°? No, wait, maybe first find ∠2. Wait, the triangle with 25°: ∠1 + 25° + 90° = 180°? Wait, no, the right angle is marked, so the triangle with 25° has angles 25°, 90°, and ∠1. So ∠1 = 180° - 90° - 25° = 65°? Wait, no, maybe another approach. Wait, the angle with 51°: in the triangle with 51° and right angle, the angle at the vertex (∠3) would be 180° - 90° - 51° = 39°? Wait, no, maybe we need to find the angle in the other triangle. Wait, the problem is to find m∠6. Let's see the right triangles and the angles given.

Wait, there's a right angle (90°) and a 28° angle, and a 35° angle? Wait, maybe we can find the angle in the triangle containing ∠6. Let's consider the sum of angles in a triangle or linear pairs.

Wait, first, let's find the angle adjacent to ∠5. Wait, in the right triangle with 28°, the angle (let's call it x) would be 90° - 28° = 62°? Wait, no. Wait, maybe we can find the angle in the triangle with 35°: 90° - 35° = 55°? Wait, no, let's look at the angles around the point. Wait, maybe the key is to find the angle in the triangle and then use complementary or supplementary angles.

Wait, let's start over. Let's look at the triangle with 25°: it's a right triangle, so the other acute angle (∠1) is 90° - 25° = 65°? Wait, no, 90 - 25 is 65. Then, the triangle with 51°: it's a right triangle, so the other acute angle (∠3) is 90° - 51° = 39°. Then, since ∠1 + ∠2 + ∠3 = 90°? Wait, no, maybe the three angles ∠1, ∠2, ∠3 are in a right angle? Wait, there's a right angle marked, so ∠1 + ∠2 + ∠3 = 90°? Wait, if ∠1 is 65° (from 90 -25), ∠3 is 39° (from 90 -51), then ∠2 = 90 - 65 -39 = -14? That can't be. So I must have made a mistake.

Wait, maybe the triangle with 25° is not a right triangle? Wait, no, there's a right angle symbol. Wait, maybe the angle 25° is in a triangle with ∠3 and another angle. Wait, perhaps the correct approach is to find the angle in the triangle containing ∠6. Let's see: there's a right angle (90°), a 28° angle, and we need to find the angle adjacent to ∠6. Wait, maybe ∠6 is complementary to some angle. Wait, let's look at the angles given: 25°, 51°, 28°, 35°, and right angles.

Wait, another approach: the sum of angles in a triangle is 180°. Let's consider the triangle where ∠6 is located. Let's see, there's a right angle (90°), a 28° angle, and a 35° angle? No, maybe not. Wait, let's find the angle in the triangle with 28°: 90° - 28° = 62°, and in the triangle with 35°: 90° - 35° = 55°? No, that doesn't help. Wait, maybe the angle ∠6 is equal to 180° - (90° + 28° + 35°)? No, that doesn't make sense.

Wait, maybe the key is to find the angle in the triangle and then use the fact that ∠6 is complementary to some angle. Wait, let's look at the angle marked 28°: in the right triangle, the other acute angle is 90° - 28° = 62°. Then, the angle with 35°: in the right triangle, the other acute angle is 90° - 35° = 55°. Then, maybe ∠6 is 180° - 62° - 55°? No, that's 63. Wait, no.

Wait, maybe I should look at the angles around the point. Wait, there's a right angle (90°), and angles 25°, 51°, 28°, 35°, and we need to find ∠6. Wait, let's calculate the angle in the triangle:

First, find the angle in the triangle with 25°: 180 - 90 -25 = 65° (∠1).

Then, the triangle with 51°: 180 - 90 -51 = 39° (∠3).

Then, since ∠1 + ∠2 + ∠3 = 90° (because it's a right angle), so 65 + ∠2 + 39 = 90 → ∠2 = 90 - 104 = -14. That's impossible. So my mistake is in assuming ∠1, ∠2, ∠3 are in a right angle. Maybe the right angle is between ∠2 and the 51° angle? Wait, the diagram has a right angle (square) between the base and the vertical line, so ∠2 is part of that right angle. Wait, maybe the triangle with 25°: angle at the top is ∠1, right angle at the bottom, so ∠1 = 90 -25 = 65. Then, the triangle with 51°: angle at the top is ∠3, right angle at the bottom, so ∠3 = 90 -51 = 39. Then, since ∠1 + ∠2 + ∠3 = 90 (because the vertical line and the two other lines form a right angle), so 65 + ∠2 + 39 = 90 → ∠2 = -14. That's wrong. So maybe the triangle with 25° is not a right triangle? Wait, the diagram shows a right angle (square) there, so it must be. Maybe the 51° is not in a right triangle? Wait, the diagram has a square (right angle) next to the 51° angle, so that triangle is right-angled. So maybe my approach is wrong.

Wait, let's look at the other side. The triangle with 28°: it's a right triangle, so the other acute angle is 90 -28 = 62. The triangle with 35°: right triangle, other acute angle is 90 -35 = 55. Then, ∠5 is 90 - 62 = 28? No, wait, ∠5 and the 28° angle: are they complementary? Wait, maybe ∠5 is 90 -28 = 62? No, 90 -28 is 62. Then, ∠6: in the triangle, the angles are 35°, 62°, and ∠6? Wait, no, the triangle with 35° and ∠6: maybe it's a right triangle? Wait, the diagram has a right angle (square) there, so 35 + ∠6 + 90 = 180? No, 35 + ∠6 + (angle) = 180. Wait, maybe the key is to find the angle in the triangle and then use the fact that ∠6 is equal to 90 - (something).

Wait, let's try a different approach. Let's find the angle in the triangle with 28°: 90 -28 = 62. Then, the angle in the triangle with 35°: 90 -35 = 55. Then, ∠6 = 180 - 62 - 55 = 63? No, that doesn't make sense. Wait, maybe the sum of angles around the point. Wait, the problem is to find m∠6. Let's look at the angles given: 25°, 51°, 28°, 35°, and right angles.

Wait, another way: in the right triangle with 28°, the angle (let's call it A) is 90° - 28° = 62°. In the right triangle with 35°, the angle (let's call it B) is 90° - 35° = 55°. Then, ∠6 = 180° - 62° - 55° = 63°? No, that's not right. Wait, maybe ∠6 is complementary to (90° - 28° - 35°)? No, 90 -28 -35 = 27, so 90 -27 = 63? Wait, 28 + 35 = 63, so 90 -63 = 27? No, that's not.

Wait, maybe I made a mistake in the initial steps. Let's look at the answer. Wait, the correct approach is:

First, find the angle in the triangle with 25°: 90° - 25° = 65° (∠1).

Then, the triangle with 51°: 90° - 51° = 39° (∠3).

Then, since ∠1 + ∠2 + ∠3 = 90° (right angle), so ∠2 = 90° - 65° - 39° = -14°, which is impossible. So clearly, my assumption is wrong. Maybe the triangle with 25° is not a right triangle? Wait, the diagram shows a right angle (square) there, so it must be. Maybe the 51° is not in a right triangle? Wait, the diagram has a square next to the 51° angle, so that triangle is right-angled. So maybe the 25° is in a different triangle. Wait, maybe the angle ∠1 is 25°, no, the 25° is at the bottom left.

Wait, let's look at the angle adjacent to ∠6. There's a right angle, a 28° angle, and a 35° angle. Wait, maybe the angle ∠6 is equal to 90° - (28° + 35°) = 27°? No, 28 +35=63, 90-63=27. But that doesn't seem right. Wait, no, maybe the angle is 180 - (90 + 28 + 35) = 27? No, 90+28+35=153, 180-153=27. Wait, that could be. Wait, 90 (right angle) +28 +35 +∠6=180? No, 90+28+35=153, so 180-153=27. So m∠6=27°? Wait, but that seems low. Wait, let's check again.

Wait, the triangle with 28°: right angle, so the other two angles are 28° and 62° (90-28=62). The triangle with 35°: right angle, so the other two angles are 35° and 55° (90-35=55). Then, the angle ∠6 is adjacent to these two angles? Wait, no, maybe ∠6 is 90° - (62° - 35°)? No, that's not. Wait, maybe the correct answer is 27°, but let's see.

Wait, another approach: the sum of angles in a triangle is 180°. Let's consider the triangle containing ∠6. It has a right angle (90°), a 28° angle, and a 35° angle? No, that would be 90+28+35=153, so the remaining angle is 27°, which is ∠6. Yes, that makes sense. So m∠6=27°.

Wait, let's verify:

In the right triangle with 28°, the angle (let's call it X) is 90° - 28° = 62°.

In the right triangle with 35°, the angle (let's call it Y) is 90° - 35° = 55°.

Then, the angle between X and Y is 62° - 55° = 7°? No, that's not. Wait, no, the correct way is:

The angle ∠6 is in a triangle where the other two angles are (90° - 28°) and (90° - 35°), but that's not. Wait, no, the triangle containing ∠6 has angles: 90° (right angle), 28° + 35° = 63°, so ∠6 = 180° - 90° - 63° = 27°. Yes, that's correct. So m∠6=27°.

Step2: Calculate m∠6

We know that in a triangle, the sum of angles is \(180^\circ\). The triangle containing \(\angle 6\) has a right angle (\(90^\circ\)) and the sum of the other two angles is \(28^\circ + 35^\circ = 63^\circ\). Thus,
\[ m\angle 6 = 180^\circ - 90^\circ - 63^\circ = 27^\circ \]

Answer:

\(27\)