Question

4 multiple choice 25 points if x = 80 solve for a. si x = 80 resolver para a. triangle with ( a^circ ) at top, ( x^circ ) at bottom left, two equal sides marked options: 50, 20, 100, 80

Explanation:

Step1: Identify triangle type

The triangle has two equal sides (marked with ticks), so it's isosceles. Thus, the two base angles are equal. So the other base angle is also \( x = 80^\circ \).

Step2: Use triangle angle sum

The sum of angles in a triangle is \( 180^\circ \). Let the vertex angle be \( a \). So \( a + x + x = 180^\circ \).
Substitute \( x = 80^\circ \): \( a + 80 + 80 = 180 \).
Simplify: \( a + 160 = 180 \).
Subtract 160: \( a = 180 - 160 = 20 \)? Wait, no, wait—wait, the vertex angle is \( a \), and the two equal angles are the base angles. Wait, maybe I mixed up. Wait, the triangle: the two equal sides are the legs, so the base angles are the ones at the base. Wait, the angle labeled \( x \) is a base angle, so the other base angle is also \( x \). Then the vertex angle is \( a \). So sum is \( a + x + x = 180 \). So \( a = 180 - 2x \). Wait, \( x = 80 \), so \( 2x = 160 \), so \( a = 180 - 160 = 20 \)? But wait, that seems small. Wait, maybe I got the angles wrong. Wait, maybe the equal sides are the ones forming the angle \( a \)? No, the ticks are on the two sides adjacent to \( x \) and the other base angle. Wait, no, the diagram: the triangle has two sides with ticks, so those two sides are equal, so the angles opposite them are equal. Wait, the side opposite \( x \) is the other equal side? Wait, no, the angle \( x \) is at the base, so the two equal sides are the legs, so the base angles are equal. Wait, maybe I made a mistake. Wait, let's re-express. In an isosceles triangle, the sum of angles is \( 180^\circ \). If two angles are equal (the base angles), then \( a + 2x = 180 \). So \( a = 180 - 2x \). Given \( x = 80 \), then \( a = 180 - 2*80 = 180 - 160 = 20 \)? But the options include 20. Wait, but let me check again. Wait, maybe the angle \( a \) is one of the equal angles? No, the diagram shows \( a \) at the top, and \( x \) at the bottom left, with the two sides from the top to the bottom having ticks. So the two sides with ticks are equal, so the angles opposite them (the bottom two angles) are equal. So the bottom two angles are both \( x \), so \( x = 80 \), so the other bottom angle is 80. Then the top angle \( a = 180 - 80 - 80 = 20 \). So that's correct.

Answer:

20 (corresponding to the option with 20)