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
Holt Mcdougal Larson Pre-algebra: Student Edition 2012
1st Edition
ISBN:9780547587776
Author:HOLT MCDOUGAL
Publisher:HOLT MCDOUGAL
Chapter9: Real Numbers And Right Triangles
Section9.3: The Pythagorean Theorem
Problem 31E
Related questions
Question
![(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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb3d7805d-01cf-4904-ad50-6452ccd792d0%2F2c4d1b49-3c87-442a-8681-69176524e2ec%2Fzr4o39v_processed.jpeg&w=3840&q=75)
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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