Page 29 - Math Course 3 (Book 2)

Properties of Squares and Rhombi

Measures of a Rhombus
Let’s Begin
Examples

Proof of Theorem 7.15 M

Example 1
L
Q N

A C
P
B
Use rhombus LMNP to find the value of y if
m∠1 = y2 – 54.

The diagonals of a rhombus are
m∠1 = 90 perpendicular.
D
E y² – 54 = 90 Substitution

1
Given: BCDE is a rhombus, m∠AEB = m∠BCD, y² = 144 Add 54 to each side.
2
and AE ≅ CE.
Prove: △ABE ≅ △CDE y = ±12 Take the square root of each side.

Proof: Because opposite angle of a rhombus are
congruent and the diagonals of a rhombus Answer The value of y can be 12 or –12.
bisects each other, ∠BEC ≅ ∠DEC ≅ ∠DCE
Use rhombus LMNP to find m∠PNL if m∠MLP = 64.
≅ ∠BCE and m∠BEC = m∠DEC = m∠DCE =
m∠BCE. Because EC bisects ∠BCD, m∠PNM = m∠MLP Opposite angles are
1
m∠BCE = m∠BCD. By substitution, congruent.
2
m∠BCE = m∠AEB and thus, ∠BCE ≅ ∠AEB m∠PNM = 64 Substitution
BE ≅ BE by the Reflexive Property and it is
The diagonals of a rhombus bisect the angles. So,
given that AE ≅ CE. 1
m∠PNL = (64) or 32.
2
Therefore, △ABE ≅ △CDE by SAS. Answer 32

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