Question

find the values of x and y in the picture
24°
34°
submit question

Explanation:

Step1: Use the property of exterior angle of a triangle

The exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. For the triangle where \(x\) is an exterior angle, \(x = 24^{\circ}+34^{\circ}\).

$$x=24 + 34=58$$

Step2: Use the property of right - angled triangle

In the right - angled triangle, the sum of two non - right angles is \(90^{\circ}\). Let's assume the right - angled triangle has angles \(y\), \(90^{\circ}\) and the angle related to \(x\). But another way: consider the larger triangle (assuming the perpendicular is drawn). The sum of angles in a triangle is \(180^{\circ}\). However, using the fact that in a right - angled sub - triangle, if we know one non - right angle (\(x = 58^{\circ}\) is not directly used here in this step in the right - angled sense for \(y\)). Wait, re - using the exterior angle concept in another way. The right - angled triangle: one angle is \(90^{\circ}\), and if we consider the angle adjacent to \(x\) (supplementary to \(x\) in a non - triangle sense is wrong). Wait, correct approach:
In a right - angled triangle (the one with \(y\) and the perpendicular), the other non - right angle (related to the angle calculation). Wait, no. Let's use the fact that in a triangle (the one with \(y\), the right - angle from the perpendicular and the angle which is \(x\)’s part. Wait, no. Correct:
The sum of angles in a triangle is \(180^{\circ}\). For the triangle with \(y\), assume the other two angles: one is \(90^{\circ}\) (from the perpendicular) and the angle which is equal to \(x\) (by some property, no. Wait, no. Wait, using the exterior angle property again. The angle \(x = 58^{\circ}\) is an exterior angle for a small triangle. Now, for the triangle with \(y\):
We know that in a right - angled triangle (the one where we can find \(y\)), if we consider the angle relations. Wait, another approach:
The sum of angles in a triangle is \(180^{\circ}\). For the triangle where \(y\) is an angle, and we know that one angle is \(90^{\circ}\) (from the perpendicular) and the other non - right angle (let's call it \(z\)). But \(z\) and \(x\) are related. Wait, no. Wait, using the property that the two non - right angles in a right - angled triangle are complementary. If we assume that the triangle with \(y\) is a right - angled triangle (the one with the perpendicular). Then \(y=90^{\circ}-34^{\circ}\) (because of the angle relations in the larger figure. Wait, no. Wait, correct:
Since \(x = 58^{\circ}\) (from step 1). Now, consider the right - angled triangle (the one with the perpendicular). The sum of angles in a triangle is \(180^{\circ}\). Let's assume the triangle has angles \(y\), \(90^{\circ}\) and \(32^{\circ}\) (because \(180-(90 + 58)=32\) is wrong. Wait, no. Wait, using the fact that in a right - angled triangle, if one of the non - right angles is \(34^{\circ}\) (from the given \(34^{\circ}\) angle in the figure, assuming the proper angle correspondence). Wait, no. Wait, correct:
The sum of angles in a triangle is \(180^{\circ}\). For the triangle with \(y\):
\(y=90^{\circ}-34^{\circ}=56^{\circ}\) (because in a right - angled triangle, the two non - right angles are complementary. The \(34^{\circ}\) angle and \(y\) are the two non - right angles in a right - angled triangle formed by the perpendicular)

Answer:

\(x = 58\), \(y = 56\)