Question

1. if two sides of a triangle are congruent. what other information do you know about this triangle?

a) the vertices opposite the sides are congruent.
b) the angles opposite the sides are congruent.
c) the measures opposite the sides are congruent
d) the sides opposite the sides are congruent.

1. if an isosceles triangle has a base angle measuring 40°, then what is the measure of the vertex angle?

a) 120°
b) 80°
c) 40°
d) 100°

Explanation:

1.

Step1: Recall the property of isosceles triangles

In an isosceles triangle (a triangle with two congruent sides), the angles opposite the congruent sides are congruent. Vertices are points, not congruent in the sense of angle - side relationships here. "Measures opposite" is an unclear term. And "the sides opposite the sides" is a redundant and incorrect description.

2.

Step1: Use the angle - sum property of triangles

The sum of the interior angles of a triangle is \(180^{\circ}\). In an isosceles triangle, the base angles are equal. Let the base angles be \(B_1 = B_2=40^{\circ}\) and the vertex angle be \(V\).

Step2: Apply the formula \(B_1 + B_2+V = 180^{\circ}\)

Substitute \(B_1 = B_2 = 40^{\circ}\) into the formula: \(40^{\circ}+40^{\circ}+V=180^{\circ}\).

Step3: Solve for \(V\)

First, simplify the left - hand side: \(80^{\circ}+V = 180^{\circ}\). Then, subtract \(80^{\circ}\) from both sides: \(V=180^{\circ}- 80^{\circ}=100^{\circ}\).

Answer:

1. B. The angles opposite the sides are congruent.
2. D. \(100^{\circ}\)