Question

1. the measure of the vertex angle of an isosceles triangle is 12 more than 5 times the measure of a base angle. determine the sum of the measures of the base angles. the sum of the measures of the base angles is type your answer...

Explanation:

Step1: Define variables

Let the measure of each base angle be \( x \). Then the sum of the base angles is \( 2x \). The vertex angle is \( 5(2x)+12 = 10x + 12 \).

Step2: Use triangle angle sum

The sum of angles in a triangle is \( 180^\circ \). So, \( 2x+(10x + 12)=180 \).

Step3: Solve the equation

Combine like terms: \( 12x+12 = 180 \). Subtract 12: \( 12x=168 \). Divide by 12: \( x = 14 \). Then the sum of base angles is \( 2x=28 \)? Wait, no, wait. Wait, the vertex angle is 12 more than 5 times the sum of the base angles. Wait, I misread. Let's redefine. Let \( S \) be the sum of the base angles. Then vertex angle \( V=5S + 12 \). And \( V+S=180 \) (since sum of angles in triangle is 180, and sum of base angles is \( S \), vertex is \( V \)). So substitute \( V \): \( 5S + 12+S=180 \). Combine: \( 6S+12 = 180 \). Subtract 12: \( 6S=168 \). Divide by 6: \( S = 28 \)? Wait, no, that can't be. Wait, no, in an isosceles triangle, the two base angles are equal, so sum of base angles is \( 2x \), vertex is \( V \). So \( V + 2x=180 \). And \( V=5(2x)+12=10x + 12 \). So \( 10x + 12+2x=180 \). \( 12x+12=180 \). \( 12x=168 \). \( x = 14 \). Then sum of base angles is \( 2x = 28 \)? But that would make vertex angle \( 10*14 +12=152 \), and 152+28=180, which works. Wait, but the problem says "the measure of the vertex angle of an isosceles triangle is 12 more than 5 times the measure of a base angle". Oh! I misread. It's 5 times the measure of a base angle, not the sum. Let's correct. Let base angle be \( x \), so two base angles: \( 2x \). Vertex angle \( V = 5x + 12 \). Then \( V + 2x=180 \). So \( 5x + 12+2x=180 \). \( 7x+12=180 \). \( 7x=168 \). \( x = 24 \). Then sum of base angles is \( 2x = 48 \). Wait, now I see the mistake. The problem says "12 more than 5 times the measure of a base angle", not the sum. So let's redo.

Step1: Correct variable definition

Let \( x \) be the measure of one base angle. Then the vertex angle \( V = 5x + 12 \). The sum of the two base angles is \( 2x \). The sum of all angles in a triangle is \( 180^\circ \), so \( V + 2x = 180 \).

Step2: Substitute \( V \)

Substitute \( V = 5x + 12 \) into the equation: \( (5x + 12) + 2x = 180 \).

Step3: Simplify and solve

Combine like terms: \( 7x + 12 = 180 \). Subtract 12 from both sides: \( 7x = 168 \). Divide both sides by 7: \( x = 24 \).

Step4: Find the sum of base angles

The sum of the two base angles is \( 2x \). Substitute \( x = 24 \): \( 2 \times 24 = 48 \)? Wait, no, wait. Wait, the problem says "Determine the sum of the measures of the base angles". Wait, no, let's check the problem again. "The measure of the vertex angle of an isosceles triangle is 12 more than 5 times the measure of a base angle. Determine the sum of the measures of the base angles."

So vertex angle \( V = 5 \times (\text{base angle}) + 12 \). Let base angle be \( x \), so \( V = 5x + 12 \). Sum of angles: \( V + x + x = 180 \) (since two base angles, each \( x \)). So \( 5x + 12 + 2x = 180 \). \( 7x + 12 = 180 \). \( 7x = 168 \). \( x = 24 \). Then sum of base angles is \( x + x = 48 \)? Wait, no, that's 2x. Wait, 2*24=48. But let's check the vertex angle: 5*24 +12=120+12=132. Then 132+48=180. Yes, that works. Wait, earlier I made a mistake in the first interpretation. The problem says "5 times the measure of a base angle", not the sum of the base angles. So that's the key. So the sum of the base angles is 2x, where x is one base angle. So solving:

Let \( x \) = measure of one base angle.

Vertex angle \( V = 5x + 12 \).

Sum of angles in triangle: \( V + x + x = 180 \)

Substitute \( V \): \( 5x + 12 + 2x = 180 \)

Combine like terms: \( 7x + 12 = 180 \)

Subtract 12: \( 7x = 168 \)

Divide by 7: \( x = 24 \)

Sum of base angles: \( x + x = 2x = 48 \)

Wait, but let's check again. The problem says "the measure of the vertex angle is 12 more than 5 times the measure of a base angle". So vertex angle = 5*(base angle) +12. So if base angle is 24, vertex is 5*24+12=132. Sum of base angles: 24+24=48. 132+48=180. Correct.

Answer:

48