Question

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1. a. what are the values of x and y in the figure?

image of a triangle with angles labeled 77°, 96°, 35°, y°, x°
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Explanation:

Step1: Find y using triangle angle sum

In the left triangle, angles sum to \(180^\circ\). So \(y + 77 + 96 = 180\).
\(y + 173 = 180\)
\(y = 180 - 173 = 7\)? Wait, no, wait. Wait, the 96° is adjacent? Wait, no, maybe the right triangle. Wait, the right triangle has angles 35°, x°, and the angle adjacent to 96°? Wait, no, the straight line: 96° and the angle adjacent to x in the right triangle: 180 - 96 = 84°? Wait, no, let's re-examine.

Wait, the left triangle: angles are 77°, 96°? No, that can't be, because 77 + 96 = 173, so y would be 7, but that seems odd. Wait, maybe the 96° is an exterior angle? Wait, no, the figure: a triangle with a segment from the top vertex to the base, splitting the base into two angles: 77° and 96°? Wait, no, the base is a straight line, so 77° + 96°? No, that's more than 180. Wait, no, the 96° is an angle in the right triangle? Wait, maybe the left triangle has angles 77°, y°, and the angle supplementary to 96°? Wait, the base is a straight line, so the angle adjacent to 96° is \(180 - 96 = 84^\circ\)? No, that doesn't make sense. Wait, maybe the right triangle: angles are 35°, x°, and the angle at the base is \(180 - 96 = 84^\circ\)? Wait, no, let's start over.

Wait, the left triangle: angles are 77°, y°, and the angle between them and the segment. Wait, the sum of angles in a triangle is 180°. So in the left triangle, angles are 77°, y°, and (180 - 96)°? Wait, 96° is adjacent to the left triangle's base angle? So the base angle of the left triangle is \(180 - 96 = 84^\circ\)? No, that's not right. Wait, maybe the 96° is an angle in the right triangle. Wait, the right triangle has angles 35°, x°, and 96°? No, 35 + 96 = 131, so x would be 49. But then the left triangle: 77°, y°, and (180 - 96) = 84°? 77 + 84 = 161, so y = 19. No, that's confusing. Wait, maybe the figure is a triangle with a height, but no. Wait, the original problem: the figure has a triangle with a segment from the top vertex to the base, creating two triangles. The left triangle has angles 77°, y°, and the angle at the top (let's call it angle A). The right triangle has angles 35°, x°, and the angle at the base adjacent to 96°? Wait, the base is a straight line, so the two base angles (77° and the angle adjacent to 96°) should add up to 180°? No, 77 + 96 = 173, which is less than 180. Wait, maybe the 96° is an angle in the right triangle, and the left triangle's base angle is 77°, and the angle between the two triangles at the base is 180 - 96 = 84°? No, this is confusing. Wait, let's use the triangle angle sum properly.

Wait, in the right triangle: angles are 35°, x°, and the angle at the base which is supplementary to 96°? Wait, no, the angle at the base of the right triangle is \(180 - 96 = 84^\circ\)? No, that's not. Wait, maybe the 96° is an exterior angle. Wait, no, let's look at the left triangle: angles are 77°, y°, and the angle between the two triangles (let's call it angle B). Then the right triangle: angles are 35°, x°, and angle B. Also, angle B + 96° = 180°, so angle B = 84°. Then in the left triangle: 77 + y + 84 = 180. So 77 + 84 = 161, so y = 19. Then in the right triangle: 35 + x + 84 = 180. 35 + 84 = 119, so x = 61. Ah, that makes sense. So:

Step1: Find angle B (supplementary to 96°)

\( \angle B = 180^\circ - 96^\circ = 84^\circ \)

Step2: Find y in left triangle

Sum of angles in triangle: \( y + 77^\circ + 84^\circ = 180^\circ \)
\( y + 161^\circ = 180^\circ \)
\( y = 180^\circ - 161^\circ = 19^\circ \)

Step3: Find x in right triangle

Sum of angles in triangle: \( x + 35^\circ + 84^\circ = 180^\circ \)
\( x + 119^\circ = 180^\circ \)
\( x = 180^\circ - 119^\circ = 61^\circ \)

Answer:

\( x = 61 \), \( y = 19 \)