Question

additional details were added for this assignment. guillermo draws three isosceles triangles. in each figure, he measures a pair of angles, as shown. which conjecture is reasonable for guillermo to make? in an isosceles triangle, two of the angles are congruent. in an isosceles triangle, two of the angles are obtuse. in an isosceles triangle, exactly one angle is always acute. in an isosceles triangle, all of the angles are congruent.

Explanation:

• An isosceles triangle has the property that two of its sides are equal. By the base - angles theorem, the angles opposite the equal sides (the base angles) are congruent.
• Looking at the given triangles: In the first triangle, two angles are \(34^{\circ}\); in the second, two angles are \(64^{\circ}\); in the third, two angles are \(73^{\circ}\).
• For the option "In an isosceles triangle, two of the angles are obtuse": An obtuse angle is greater than \(90^{\circ}\). If two angles of a triangle were obtuse (\(>90^{\circ}\) each), then the sum of those two angles would be \(> 180^{\circ}\), but the sum of all three angles of a triangle is \(180^{\circ}\). So this option is incorrect.
• For the option "In an isosceles triangle, exactly one angle is always acute": In an equilateral triangle (a special case of an isosceles triangle where all sides and angles are equal), all angles are acute (\(60^{\circ}\)). So this option is incorrect.
• For the option "In an isosceles triangle, all of the angles are congruent": This is only true for equilateral triangles (a subset of isosceles triangles). But not all isosceles triangles have all angles congruent. For example, the first given triangle has angles \(34^{\circ},34^{\circ},112^{\circ}\) (since \(180-(34 + 34)=112\)).

Answer:

In an isosceles triangle, two of the angles are congruent.