Question

what must be marked on the triangles below for them to be congruent by sas?
what is the value of x?
how can you verify that your answer for the problem above is correct?
what is the value of x?

1. prove: the measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles.

Explanation:

What is the value of \(x\)?

Step1: Use the triangle angle - sum theorem

The sum of the interior angles of a triangle is \(180^{\circ}\). So, \(2x+(x + 12)+96=180\).

Step2: Simplify the left - hand side of the equation

Combine like terms: \(2x+x+12 + 96=3x+108\). So, \(3x+108 = 180\).

Step3: Solve for \(x\)

Subtract \(108\) from both sides: \(3x=180 - 108=72\). Then divide both sides by \(3\): \(x=\frac{72}{3}=24\).

Substitute \(x = 24\) into the expressions for the angles of the triangle.
The angles are \(2x=2\times24 = 48^{\circ}\), \(x + 12=24+12=36^{\circ}\), and \(96^{\circ}\).
Check if their sum is \(180^{\circ}\): \(48+36 + 96=180^{\circ}\).

Step1: Use the exterior angle theorem

The exterior angle theorem states that an exterior angle of a triangle is equal to the sum of the two non - adjacent (remote) interior angles.
Here, the exterior angle \(x\) is equal to \(28^{\circ}+58^{\circ}\).

Answer:

\(x = 24\)

How can you verify that your answer for the problem above is correct?