This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Solve Question 1. The question states that triangle ABC is congruent to triangle PQR ( ABC PQR). When two triangles are congruent, their corresponding sides and corresponding angles are equal. The order of the vertices in the congruence statement indicates which parts correspond. A corresponds to P. B corresponds to Q. C corresponds to R. We are given that BAC = QPR. This is consistent with the congruence statement, as BAC is the angle at vertex A, and QPR is the angle at vertex P. The question asks which side BC is equal to in triangle PQR. Since B corresponds to Q and C corresponds to R, the side BC in ABC corresponds to side QR in PQR. Therefore, side BC is equal to side QR. The answer is QR. Step 2: Solve Question 2. The image shows two triangles, PQR and TUV. We need to show that they are congruent. Let's examine the markings on the triangles: In PQR: • Side PQ has two tick marks. • Side PR has one tick mark. • Angle Q is marked with a right angle symbol (90^). In TUV: • Side TU has two tick marks. • Side TV has one tick mark. • Angle U is marked with a right angle symbol (90^). Comparing the corresponding parts: 1. Side PQ has two tick marks, and side TU has two tick marks. So, PQ = TU. 2. Angle Q is a right angle, and angle U is a right angle. So, Q = U = 90^. 3. Side PR has one tick mark, and side TV has one tick mark. So, PR = TV. We have two sides and a non-included angle (SSA). However, for right-angled triangles, if the hypotenuse and one leg are equal, the triangles are congruent by the Hypotenuse-Leg (HL) congruence theorem. In PQR, PR is the hypotenuse (opposite the right angle Q). PQ is a leg. In TUV, TV is the hypotenuse (opposite the right angle U). TU is a leg. We have: • Hypotenuse PR = TV (both have one tick mark). • Leg PQ = TU (both have two tick marks). • Both triangles are right-angled at Q and U respectively. Therefore, PQR TUV by the Hypotenuse-Leg (HL) congruence theorem. The answer is HL Congruence Theorem. Step 3: Solve Question 3. The question states that ABE is congruent to BDC ( ABE BDC). We are given the following lengths: AE = 5 cm EB = 3 cm AB = 7 cm We need to find the measure of side BC. Since the triangles are congruent, their corresponding sides are equal. From the congruence statement ABE BDC: • Side AB corresponds to side BD. So, AB = BD. • Side BE corresponds to side DC. So, BE = DC. • Side AE corresponds to side BC. So, AE = BC. We are given AE = 5 cm. Since AE = BC, then BC = 5 cm. The measure of side BC is 5 cm. Got more? Send 'em.