Question 1: An isosceles triangle has two equal sides measuring 7 cm each. The base angles are equal. If one base angle measures 40°, find the third angle of the triangle.

Answer: 100

Question 2: In an isosceles triangle, the two equal angles are each 65°. What is the measure of the vertex angle?

Answer: 50

Question 3: Construct an isosceles triangle where each of the equal sides is 6 cm, and the vertex angle is 40°. Use the grid to draw and label the triangle.

Answer:

Question 4: An isosceles triangle has a base of 10 cm and the equal sides are 13 cm each. Calculate the height of the triangle.

Answer: 12

Question 1: A triangle is isosceles with sides measuring 9 cm and 9 cm. The base angle is 70°. Find the measure of the other two angles and explain your reasoning.

Answer: The other two angles each measure 50°. Since the sum of angles in a triangle is 180°, and the base angles are equal, then 2x + 70° = 180°, so x = 50°.

Question 2: A student claims that in an isosceles triangle, the two equal angles are always opposite the equal sides. Is this true? Justify your answer.

Answer: Yes, in an isosceles triangle, the angles opposite the equal sides are equal. This is a property of isosceles triangles.

Question 3: Given an isosceles triangle with equal sides of 8 cm and a vertex angle of 60°, find the length of the base.

Answer: Approx. 13.86 cm

Question 1: A roof truss in a building forms an isosceles triangle with the two equal sides being 5 meters each. The vertex angle is 45°. Calculate the height of the roof at the apex.

Answer: 2.5

Question 2: A farmer designs an isosceles triangular fence with a base of 12 meters and equal sides of 15 meters. How much fencing material is needed in total?

Answer: 42

Question 1: Construct an isosceles triangle such that the base is 6 cm, each of the equal sides is 10 cm, and the vertex angle is 30°. Verify your construction and calculate the area of the triangle.

Answer: Approx. 14.64 cm²

Question 2: Prove that the base angles of an isosceles triangle with vertex angle 20° are equal, and find their measure.

Answer: 80° each

Question 1: A student draws an isosceles triangle and claims the base angles are each 60°, and the vertex angle is 60°. Is this correct? Identify the mistake and correct it.

Answer: Incorrect. All angles are 60°, forming an equilateral triangle, which is a special case of isosceles. No mistake in the angles; however, the student should recognize this special case.

Question 2: A student states that in an isosceles triangle, the unequal side is always opposite the largest angle. Is this true? Explain.

Answer: Yes, in any triangle, the side opposite the largest angle is the longest. In an isosceles triangle, the unequal side is opposite the vertex angle if the triangle is scalene, but if it's strictly isosceles, the unequal side does not exist; this statement only applies when the triangle is scalene.