Question

for questions 13 and 14, find the value of x.
13.
(8x - 23)°
34°
x =
14.
(9x - 3)

1. δcde is an isosceles triangle with ∠d ≅ ∠e. if cd = 4x +

find x and the measure of each side.

Explanation:

Step1: Identify triangle type

The triangle is isosceles (two equal sides), so the base angles are equal. Thus, \((8x - 23)^\circ = 34^\circ\)? Wait, no—wait, in an isosceles triangle, the angles opposite equal sides are equal. Wait, the marked sides are the legs, so the base angles (the ones at the equal sides) are equal? Wait, no, the vertex angle is \(34^\circ\), and the two base angles are equal. Wait, no, the angle marked \((8x - 23)^\circ\) and the other base angle (not shown) are equal? Wait, no, the triangle has two equal sides (marked with ticks), so it's isosceles with the two equal sides as the legs, so the base angles (the angles at the base) are equal? Wait, no, the vertex angle is the one at the bottom (34°), and the two upper angles are equal. So the two base angles (the ones at the top) are equal. Wait, no, in an isosceles triangle, the angles opposite the equal sides are equal. So if the two sides with ticks are the legs, then the angles opposite them (the base angles) are equal. Wait, maybe I got it wrong. Let's re-express: in an isosceles triangle, the sum of angles is \(180^\circ\). Let the two equal angles be \((8x - 23)^\circ\) each, and the vertex angle is \(34^\circ\). So sum of angles: \(2(8x - 23) + 34 = 180\)? Wait, no—wait, maybe the angle \((8x - 23)^\circ\) is equal to \(34^\circ\)? Wait, no, that can't be. Wait, maybe the triangle has the two equal angles as \((8x - 23)^\circ\) and \(34^\circ\)? No, the ticks on the sides mean the sides are equal, so the angles opposite are equal. So if the two sides with ticks are the legs, then the angles opposite them (the base angles) are equal. Wait, the angle marked \((8x - 23)^\circ\) and the other angle (not labeled) are equal, and the vertex angle is \(34^\circ\). So sum of angles: \((8x - 23) + (8x - 23) + 34 = 180\)? Wait, no, that would be if the two equal angles are \((8x - 23)^\circ\) and the vertex is \(34^\circ\). Wait, let's check:

Wait, maybe I made a mistake. Let's start over. The triangle has two equal sides (marked), so it's isosceles. Therefore, the angles opposite those sides are equal. So the angle labeled \((8x - 23)^\circ\) and the angle opposite (the other base angle) are equal. The vertex angle (the one at the bottom) is \(34^\circ\). So sum of angles: \(2(8x - 23) + 34 = 180\). Let's solve that:

Step2: Solve for x

\(2(8x - 23) + 34 = 180\)

First, expand: \(16x - 46 + 34 = 180\)

Simplify: \(16x - 12 = 180\)

Add 12 to both sides: \(16x = 192\)

Divide by 16: \(x = 12\)

Wait, but let's check: if \(x = 12\), then \(8x - 23 = 96 - 23 = 73\). Then two angles of 73° and one of 34°: \(73 + 73 + 34 = 180\). Yes, that works. So the correct equation is \(2(8x - 23) + 34 = 180\), leading to \(x = 12\).

Answer:

\(x = 12\)