Q33 Prove Theorem 5-14 for one diago... [FREE SOLUTION]

91影视

Geometry

Ray C. Jurgensen, Richard G. Brown, John W. Jurgensen

$Math Studyset 91影视 Explanations$ Math

Student Edition 路 227 pages

ISBN: 9780395977279

Chapter 5: Q33 (page 188)

Prove Theorem 5-14 for one diagonal of the rhombus, (Note that a proof for the other would be similar, step-by-step).

Short Answer

Expert verified

Theorem 5-14 is proved by the definition of a rhombus, SSS postulate, parts of the congruent triangle are congruent, the definition of an angle bisector, the definition of a rhombus.

Step by step solution

Step 1. State the theorem 5-14.

The theorem is:

For one diagonal of a rhombus bisects two angles of the rhombus.

Prove, $\overset{炉}{R T}$ bisects $鈭� Q R S$ and $鈭� Q T S$ .

And prove $\overset{炉}{Q S}$ bisects $鈭� T Q R$ and $鈭� T S R$ .

Step 2. State the proof.

The given statement can be proved as below:

Statement	Reasons
Rhombus $Q R S T$ , $\overset{炉}{Q S}$ and $\overset{炉}{R T}$ are diagonals	Given.
$\overset{炉}{Q R} 鈮� \overset{炉}{Q T} 鈮� \overset{炉}{T S} 鈮� \overset{炉}{S R}$	Definition of a rhombus.
In $螖 Q R S$ and $螖 Q T S$ , $\overset{炉}{Q R} 鈮� \overset{炉}{Q T}$ , $\overset{炉}{R S} 鈮� \overset{炉}{T S}$ and $\overset{炉}{Q S} 鈮� \overset{炉}{Q S}$	Proved above.
$螖 Q R S 鈮� 螖 Q T S$ .	SSS postulate.
$鈭� T Q S 鈮� 鈭� R Q S$ , $鈭� T S Q 鈮� 鈭� T S R$	Definition of an angle bisector.
$\overset{炉}{Q R} 鈮� \overset{炉}{Q T} 鈮� \overset{炉}{T S} 鈮� \overset{炉}{S R}$	Definition of a rhombus.
In $螖 T Q R$ and $螖 T S R$ , $\overset{炉}{T Q} 鈮� \overset{炉}{T S}$ , $\overset{炉}{Q R} 鈮� \overset{炉}{R S}$ and $\overset{炉}{T R} 鈮� \overset{炉}{T R}$ .	Proved above.
$螖 T Q R 鈮� 螖 T S R$ .	SSS postulate.
$鈭� Q T R 鈮� 鈭� S T R$ , $鈭� Q R T 鈮� 鈭� S R T$	Parts of congruent triangle are congruent.
$\overset{炉}{R T}$ bisects $鈭� Q R S$ and $鈭� Q T S$	Definition of an angle bisector.

Step 3. State the conclusion.

Therefore, Theorem 5-14 is proved by definition of a rhombus, SSS postulate, parts of the congruent triangle are congruent, the definition of an angle bisector, definition of a rhombus.