Download Ch. 7 Review Guide/Notes

Name _______________________________ Period ____________ Date __________________
Ch. 7 Review Packet Reteaching 7-1
Introduction to Geometry Pairs of Angles
•
Vertical angles are pairs of opposite angles formed by two intersecting lines. They are
congruent.
Example 1: < 1 and < 3, < 4 and < 2
•
Adjacent angles have a common vertex and a common side, but no common interior points.
Example 2: < 1 and < 2, < 1 and < 4
•
Two angles who sum to 180° are supplementary.
Example 3: < 1 and < 4 are supplementary angles. < 3 is also a supplement of <4.
* If you know the measure of one supplementary angle, you can find the measure of the other. *
If m < 4 is 120°, then m < 1 is 180° – 120°, or 60°.
•
Two angles who sum to 90° are complementary.
Example 4: < 5 and < 6 are complementary angles. < 6 is a complement of < 5.
*If you know the measure of one complementary angle, you can find the measure of the other.*
If m < 5 is 30°, then m < 6 is 90° – 30°, or 60°.
Name _______________________________ Period ____________ Date __________________
Ch. 7 Review Packet Reteaching 7-2
Introduction to Geometry Angles and Parallel Lines Look at the figure at the right.
•
suur
suur suur suur
Line AB is parallel to line CD (AB P CD )
•
suur
Line EF is a transversal.
Alternate interior angles lie within a pair of lines and on opposite sides of the transversal.
Example 1: < 3 and < 5, < 4 and < 6.
Alternate interior angles are congruent. If m < 4 is 60°, then m < 6 is also 60°.
Corresponding angles lie on the same side of the transversal and in corresponding positions.
Example 2: < 1 and < 5, < 3 and < 7.
Corresponding angles are congruent. If m < 1 is 120°, then m < 5 is also 120°.
Supplementary angles are any two angles that sum to 180°.
Example 3: < 1 and < 2, < 3 and < 8.
If m < 1 is 120°, then its supplement (m < 2 ) is 180 – 120 = 60°.
Complimentary angles are any two angles that sum to 90°.
If m < 2 is 60°, then its compliment is 90 – 60 = 30°.
Name _______________________________ Period ____________ Date __________________
Ch. 7 Review Packet Reteaching 7-3
Introduction to Geometry Congruent Figures
Congruence statements reveal corresponding parts.
Example 1 AB corresponds to DE . < C corresponds to < F.
Corresponding parts are congruent ( ≅ ).
Example 2: AB ≅ DE < C ≅ < F
ΔABC ≅ ΔDEF
Triangles are congruent if you can show just three parts are congruent.
Side-Side-Side (SSS) (The marks show which parts are congruent.)
Side-Angle-Side (SAS) (The arcs show which angels are congruent.)
Angle-Side-Angle (ASA)
Name _______________________________ Period ____________ Date __________________
Ch. 7 Review Packet Introduction to Geometry Reteaching 7-4
Similar Figures
Similar polygons have congruent corresponding angles and corresponding sides that are in
proportion. The symbol ~ means is similar to.
Example: Is parallelogram ABCD ~ parallelogram KLMN?
1
Check corresponding angles.
< A ≅ < K, < B ≅ < L, < C ≅ < M, and < D ≅ < N
2
Compare corresponding sides.
AB 8 2 BC 12 2
= =
=
=
KL 4 1 LM
6 1
CD 8 2 DA 12 2
= =
=
=
MN 4 1 NK
6 1
Corresponding angles are congruent. Corresponding sides are in proportion. The
parallelograms are similar.
You can use proportions to find unknown lengths in similar figures.
1
To find EF, use a proportion.
2
Substitute.
3 Use cross products.
4 Solve.
EF = 5
AB BC
=
DE EF
12 10
=
6
n
12n = 60
n=5
ΔABC ∼ ΔDEF
Name _______________________________ Period ____________ Date __________________
Ch. 7 Review Packet Reteaching 7-5
Introduction to Geometry Proving Triangles Similar
The angles of a triangle add to 180°.
You can use the angle sum to find a missing angle measure.
m∠Q + m∠R + m∠S = 180o
o
o
o
46 + m∠R + 46 = 180
92o + m∠R = 180o
92o − 92o + m∠R = 180o − 92o
m∠R = 88o
ß Angle sum.
ß Substitute.
ß Simplify.
ß Subtract.
ß Simplify.
You can also use angle measures to show that two triangles are similar.
Step 1: Use the angle sum to find m < L.
90o + 58o + m∠L = 180o
148o + m∠L = 180o
148o −148o + m∠L = 180o −148o
m∠L = 32o
Step 2: Use AA similarity.
m∠K ≅ m∠D
m∠L ≅ m∠D
ΔKLM ∼ ΔDEF
Name _______________________________ Period ____________ Date __________________
Ch. 7 Review Packet Reteaching 7-6
Introduction to Geometry Angles and Polygons
For a polygon with n sides, the sum of the measures of the interior angles is (n – 2)180°.
Example 1: A quadrilateral is a 4-sided polygon. The sum of the angle measures is:
(4 − 2) ×180
o
= 2 ×180o
= 360°
m < 1 + m < 2 + m < 3 + m < 4= 360°
Example 2: A heptagon has 7 sides.
(n − 2)180o
(7 − 2)180o
5 × 180o = 900o
The sum of the measures of the interior angles of a heptagon is 900°.
** You can divide by the number of interior angles to find the measure of each angle for a
REGULAR polygon.
900o ÷ 7 = 128.6o
Each angle in a regular heptagon has a measure of 128.6°.
An exterior angle of a polygon is an angle formed by a side and an extension of an adjacent
side.
The measure of an exterior angle of a triangle is equal to the sum of the measures of the interior
angles at the other two vertices.
m∠2 = m∠A + m∠B
= 60o + 90o
= 150o
ß Exterior angle of triangle
ß Substitute
¬Simplify