Question

samantha measured two of the angles in △pqr and found that they had measures of 65° and 70°. then, she measured two of the angles in △xyz and found that they had measures of 65° and 45°. what statement best describes the two triangles?

the two triangles cannot be congruent because the angle measures are not the same.

the two triangles are congruent because the angle measures in the two triangles are the same.

the two triangles may be congruent, but additional information is needed about the third angle in each triangle.

the two triangles may be congruent, but additional information is needed about the sides of each triangle.

Explanation:

Step1: Calculate the third angle of $\triangle PQR$

The sum of angles in a triangle is $180^{\circ}$. Let the third angle of $\triangle PQR$ be $x$. Then $x = 180-(65 + 70)=180 - 135=45^{\circ}$

Step2: Calculate the third angle of $\triangle XYZ$

Let the third angle of $\triangle XYZ$ be $y$. Then $y=180-(65 + 45)=180 - 110 = 70^{\circ}$

Step3: Analyze congruence conditions

We know that for two triangles to be congruent, we need information about side - side - side (SSS), side - angle - side (SAS), angle - side - angle (ASA), or hypotenuse - leg (HL) for right - angled triangles. Just knowing the angle measures (AAA - angle - angle - angle) is not sufficient to prove congruence.

Answer:

The two triangles may be congruent, but additional information is needed about the sides of each triangle.