Question

AABC represents a randomly drawn triangle. ABCD is an isosceles triangle with sides BC and BD congruent. Point D lies on the line passing through points A and B. Use thisinformation to answer the questions that follow.

a. As the construction shows, the measure of ZACD is greater than the measure of ZBCD. Based on this information, create an inequality involving sides AD and AC and explain your reasoning.

b. Create a sum involving side lengths from AABC that is equal to the length of line segment AD. Explain your reasoning.

Answer 1

The inequality involving sides AD and AC is AD > AC based on the angle-side relationship in triangles. The sum involving side lengths from ΔABC that equals the length of segment AD is AD = AB + BC, recognizing that BD and BC are congruent in the isosceles triangle ABCD.

Step-by-step explanation:

To address part (a) of the question, we must apply the concept that in any triangle, larger angles are opposite longer sides. Given that ∠ACD is greater than ∠BCD, we can deduce that side AD is longer than side AC, as they are opposite these angles respectively. Therefore, we can write the inequality AD > AC. Our reasoning is based on the triangle inequality theorem which states that in any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side.

For part (b), to express the length of line segment AD as a sum involving sides from ΔABC, we should consider that AD consists of AB + BD. Given that triangle ABCD is isosceles with BC ≅ BD, we can infer that AB + BC equals the length of AD, i.e., AD = AB + BC. This is justified by the property that in isosceles triangles, the lengths of the congruent sides (BC and BD) are equal.