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NAME ______________________________________________ DATE
6-5
____________ PERIOD _____
Lesson Reading Guide
Rhombi and Squares
Get Ready for the Lesson
Read the introduction to Lesson 6-5 in your textbook.
If you draw a diagonal on the surface of one of the square wheels shown in the
picture in your textbook, how can you describe the two triangles that are formed?
Read the Lesson
1. Sketch each of the following.
a. a quadrilateral with perpendicular
diagonals that is not a rhombus
b. a quadrilateral with congruent
diagonals that is not a rectangle
c. a quadrilateral whose diagonals are
perpendicular and bisect each other,
but are not congruent
a. The diagonals are congruent.
b. Opposite sides are congruent.
c. The diagonals are perpendicular.
d. Consecutive angles are supplementary.
e. The quadrilateral is equilateral.
f. The quadrilateral is equiangular.
g. The diagonals are perpendicular and congruent.
h. A pair of opposite sides is both parallel and congruent.
3. What is the common name for a regular quadrilateral? Explain your answer.
Remember What You Learned
4. A good way to remember something is to explain it to someone else. Suppose that your
classmate Luis is having trouble remembering which of the properties he has learned in
this chapter apply to squares. How can you help him?
Chapter 6
35
Glencoe Geometry
Lesson 6-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2. List all of the following special quadrilaterals that have each listed property:
parallelogram, rectangle, rhombus, square.
NAME ______________________________________________ DATE
6-5
____________ PERIOD _____
Study Guide and Intervention
Rhombi and Squares
Properties of Rhombi A rhombus is a quadrilateral with four
congruent sides. Opposite sides are congruent, so a rhombus is also a
parallelogram and has all of the properties of a parallelogram. Rhombi
also have the following properties.
R
H
B
M
The diagonals are perpendicular.
苶
MH
苶⊥苶
RO
苶
Each diagonal bisects a pair of opposite angles.
苶
MH
苶 bisects ⬔RMO and ⬔RHO.
苶O
R
苶 bisects ⬔MRH and ⬔MOH.
If the diagonals of a parallelogram are
perpendicular, then the figure is a rhombus.
苶O
苶⊥苶
MH
苶,
If RHOM is a parallelogram and R
then RHOM is a rhombus.
O
Example
In rhombus ABCD, m⬔BAC ⫽ 32. Find the
measure of each numbered angle.
ABCD is a rhombus, so the diagonals are perpendicular and 䉭ABE
is a right triangle. Thus m⬔4 ⫽ 90 and m⬔1 ⫽ 90 ⫺ 32 or 58. The
diagonals in a rhombus bisect the vertex angles, so m⬔1 ⫽ m⬔2.
Thus, m⬔2 ⫽ 58.
B
12
32⬚ 4
A
E
3
C
D
A rhombus is a parallelogram, so the opposite sides are parallel. ⬔BAC and ⬔3 are
alternate interior angles for parallel lines, so m⬔3 ⫽ 32.
Exercises
ABCD is a rhombus.
B
C
1. If m⬔ABD ⫽ 60, find m⬔BDC.
2. If AE ⫽ 8, find AC.
3. If AB ⫽ 26 and BD ⫽ 20, find AE.
4. Find m⬔CEB.
5. If m⬔CBD ⫽ 58, find m⬔ACB.
6. If AE ⫽ 3x ⫺ 1 and AC ⫽ 16, find x.
A
D
7. If m⬔CDB ⫽ 6y and m⬔ACB ⫽ 2y ⫹ 10, find y.
8. If AD ⫽ 2x ⫹ 4 and CD ⫽ 4x ⫺ 4, find x.
9. a. What is the midpoint of 苶
FH
苶?
y
J
H
苶?
b. What is the midpoint of 苶
GJ
c. What kind of figure is FGHJ? Explain.
F
O
苶H
苶?
d. What is the slope of F
G
x
e. What is the slope of G
苶J
苶?
f. Based on parts c, d, and e, what kind of figure is FGHJ ? Explain.
Chapter 6
36
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
E
NAME ______________________________________________ DATE
6-5
Study Guide and Intervention
____________ PERIOD _____
(continued)
Rhombi and Squares
Properties of Squares
A square has all the properties of a rhombus and all the
properties of a rectangle.
Example
Find the measure of each numbered angle of
B
square ABCD.
C
1
Using properties of rhombi and rectangles, the diagonals are perpendicular
and congruent. 䉭ABE is a right triangle, so m⬔1 ⫽ m⬔2 ⫽ 90.
A
E
5
2
4
3
D
Each vertex angle is a right angle and the diagonals bisect the vertex angles,
so m⬔3 ⫽ m⬔4 ⫽ m⬔5 ⫽ 45.
Exercises
Determine whether the given vertices represent a parallelogram, rectangle,
rhombus, or square. Explain your reasoning.
1. A(0, 2), B(2, 4), C(4, 2), D(2, 0)
3. A(⫺2, ⫺1), B(0, 2), C(2, ⫺1), D(0, ⫺4)
4. A(⫺3, 0), B(⫺1, 3), C(5, ⫺1), D(3, ⫺4)
5. S(⫺1, 4), T(3, 2), U(1, ⫺2), V(⫺3, 0)
6. F(⫺1, 0), G(1, 3), H(4, 1), I(2, ⫺2)
7. Square RSTU has vertices R(⫺3, ⫺1), S(⫺1, 2), and T(2, 0). Find the coordinates of
vertex U.
Chapter 6
37
Glencoe Geometry
Lesson 6-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
2. D(⫺2, 1), E(⫺1, 3), F(3, 1), G(2, ⫺1)