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Theorem 1. If a parallelogram has a right angle, then it has four right angles and the parallelogram is a rectangle. aongruent

Theorem 1. If a parallelogram has a right angle, then it has four right angles and the parallelogram is a rectangle. aongruent

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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(Rectangle, Rhombus, and Square) The Rectangle A rectangle is a parallelogram with four right angles. Reminder: A rectangle is a parallelogram. Thus, it has all the properties of a parallelogram plus some properties of its own. Theorem 1. If a parallelogram has a right angle, then it has four right angles and the parallelogram is a rectangle. Theorem 2. The diagonals of a rectangle are congruent. Example 1. Complete a proof in theorem 2. GIVEN: Rectangle MNQP with diagonals MP and NQ. PROVE: MP = NQ Q PROOF STATEMENT REASON 1. Rectangle MNQP with diagonals MP and NQ 1. Given 2. By definition, a rectangle is a parallelogram with four right angles 3. Opposite sides of a parallelogram are congruent 2. MNQP is a parallelogram 3. MN = QP 4. MQ = MQ 4. Identity 5. ZNMQ and ZPQM are right angles 5. By Theorem 1, the four angles of a rectangle are right angles 6. All right angles are congruent 6. ZNMQ = LPQM 7. ΔΝΜΟ e ΔΡΟΜ 7. SAS Congruence Postulate 8. MP = NQ 8. Congruent Parts of Congruent Triangles are Congruent (CPCTC) The Rhombus A rhombus is a parallelogram with all four sides are congruent. Theorem 3: The diagonals of a rhombus are perpendicular. In Rhombus ROSE, MZRTO = 90° %3D MZRTE = 90° MZOTS = 90° MZETS = 90° Theorem 4: Each diagonal of a rhombus bisects opposite angles. In Rhombus ROSE, 21 2; 3 =4 E 25 = 26; 7 8 S, P.
Transcribed Image Text:(Rectangle, Rhombus, and Square) The Rectangle A rectangle is a parallelogram with four right angles. Reminder: A rectangle is a parallelogram. Thus, it has all the properties of a parallelogram plus some properties of its own. Theorem 1. If a parallelogram has a right angle, then it has four right angles and the parallelogram is a rectangle. Theorem 2. The diagonals of a rectangle are congruent. Example 1. Complete a proof in theorem 2. GIVEN: Rectangle MNQP with diagonals MP and NQ. PROVE: MP = NQ Q PROOF STATEMENT REASON 1. Rectangle MNQP with diagonals MP and NQ 1. Given 2. By definition, a rectangle is a parallelogram with four right angles 3. Opposite sides of a parallelogram are congruent 2. MNQP is a parallelogram 3. MN = QP 4. MQ = MQ 4. Identity 5. ZNMQ and ZPQM are right angles 5. By Theorem 1, the four angles of a rectangle are right angles 6. All right angles are congruent 6. ZNMQ = LPQM 7. ΔΝΜΟ e ΔΡΟΜ 7. SAS Congruence Postulate 8. MP = NQ 8. Congruent Parts of Congruent Triangles are Congruent (CPCTC) The Rhombus A rhombus is a parallelogram with all four sides are congruent. Theorem 3: The diagonals of a rhombus are perpendicular. In Rhombus ROSE, MZRTO = 90° %3D MZRTE = 90° MZOTS = 90° MZETS = 90° Theorem 4: Each diagonal of a rhombus bisects opposite angles. In Rhombus ROSE, 21 2; 3 =4 E 25 = 26; 7 8 S, P.
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