Angles Main Packet
Angles Main Packet
Angles Main Packet
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<strong>Angles</strong> page #1<br />
1. Angle: two rays with the same endpoint. The endpoint is called the vertex.<br />
An angle is named by the vertex: ∠A<br />
Vertex<br />
A<br />
X<br />
An angle can also be named by three letters with<br />
the vertex in the middle position.<br />
Y<br />
Z<br />
∠XYZ or ∠ZYX<br />
2. There are several ways to name an angle.<br />
B<br />
∠2 A number inside the angle.<br />
∠A The point at the vertex.<br />
A<br />
2<br />
C<br />
∠BAC or ∠CAB Three letters - middle letter names<br />
the vertex, other letters name a point from each<br />
side.<br />
Problems:<br />
1.<br />
K<br />
1. Name the vertex of the angle.<br />
L<br />
3<br />
J<br />
2. What rays are the sides of the angle?<br />
3. Give three other names for ∠LJK.<br />
2. Three angles with their vertices at L are shown. The name ∠L cannot be used<br />
because it would not be clear which angle was meant. Name the following angles with<br />
three letters:<br />
∠1 = __________<br />
∠2 = __________<br />
The remaining angle: __________<br />
O<br />
N<br />
2<br />
1<br />
M<br />
L
The Protractor: A protractor is an instrument used to measure angles.<br />
Problems:<br />
1. Measure the following angles:<br />
A. B.<br />
<strong>Angles</strong> page #2<br />
C. D.<br />
2. Draw angles with the measures given:<br />
A. 42° B. 133°
3. Measure angles A through G.<br />
<strong>Angles</strong> page #3<br />
C<br />
A<br />
B<br />
∠A + ∠B + ∠C = _____________________________<br />
E<br />
F<br />
D<br />
G<br />
∠D + ∠E + ∠F + ∠G = ____________________________
Notes, Notes, Notes<br />
1. A. Acute Angle: Measure is less than 90°.<br />
<strong>Angles</strong> page #4<br />
Draw an acute angle:<br />
B. Right Angle: Measure is 90°.<br />
Draw a right angle:<br />
C. Obtuse Angle: Measure is between 90° and 180°.<br />
Draw an obtuse angle:<br />
D. Straight Angle: Measure is 180°.<br />
Draw a straight angle:<br />
2. Adjacent <strong>Angles</strong>: Two angles in a plane that have a common vertex and a common<br />
side but no common interior points.<br />
∠1 and ∠2 are adjacent angles<br />
∠3 and ∠4 are not adjacent angles<br />
2 1<br />
1<br />
2<br />
3<br />
4<br />
3<br />
4<br />
3<br />
4
Problems:<br />
<strong>Angles</strong> page #5<br />
1.<br />
C<br />
D<br />
A<br />
B<br />
E<br />
A. Name 4 different pairs of adjacent angles.<br />
__________ and __________<br />
__________ and __________<br />
__________ and __________<br />
__________ and __________<br />
B. Name two different acute angles. __________ and __________<br />
C. Name one obtuse angle. __________<br />
D. Name two different right angles. __________ and __________<br />
3. Perpendicular Lines (⊥ lines): Two lines that form equal adjacent (90°) angles.<br />
Draw two lines that are perpendicular:<br />
4. Complementary angles are two angles whose measures have the sum 90°. Each<br />
angle is called a complement of the other.<br />
Draw two angles that are complementary:
5. Supplementary angles are two angles whose measures have the sum 180°. Each<br />
angle is called a supplement of the other.<br />
Draw two angles that are supplementary:<br />
<strong>Angles</strong> page #6<br />
6. <strong>Angles</strong> of a Linear Pair:<br />
A. The exterior sides of a linear pair of angles form a straight angle.<br />
D<br />
A B C<br />
∠ABD and ∠DBC are a linear pair, the measure of ∠ABC is 180°.<br />
∠ABD and ∠DBC are supplementary<br />
∠ABD +∠DBC = 180°<br />
Problems:<br />
1.<br />
∠1 + ∠2 + ∠3 = _________________<br />
1<br />
2<br />
3
2.<br />
<strong>Angles</strong> page #7<br />
C<br />
D<br />
A<br />
B<br />
E<br />
A. Name 2 different pairs of supplementary angles.<br />
__________ and __________<br />
__________ and __________<br />
B. If CB is perpendicular to AE:<br />
1. Name two different right angles. __________ and __________<br />
2. Name a pair of complementary angles. _________ and __________<br />
3. The measure of an angle is 8 times the measure of its supplement. Find the measure<br />
of each angle.<br />
4. The measure of an angle is 5 times the measure of its supplement. Find the measure<br />
of each angle.
<strong>Angles</strong> page #8<br />
5. The measure of an angle is 2 times the measure of its complement. find the<br />
measure of each angle.<br />
6. The measure of an angle is 5 more than 4 times the measure of its supplement. find<br />
the measure of each angle.<br />
7. The measure of an angle is 24 more than the measure of its complement. Find the<br />
measure of each angle.<br />
8. The measure of an angle is 20 less than 4 times the measure of its supplement. Find<br />
the measure of each angle.
<strong>Angles</strong> page #9<br />
9. Two angles are equal and complementary. What is the measure of each?<br />
10. Use your protractor to draw the following:<br />
A. Two adjacent complementary angles, one that has the measure 42°.<br />
B. Two adjacent supplementary angles, one that has the measure 42°.
11.<br />
<strong>Angles</strong> page #10<br />
1 2<br />
A. Write a formula (equation) for∠1 and ∠2: ___________________________<br />
B. If ∠1 = 2x+3 and ∠2 = x-6, then: x = __________<br />
∠1 = __________<br />
∠2 = __________<br />
C. If ∠1 = 3x-4 and ∠2 = 2x+9, then:<br />
x = __________<br />
∠1 = __________<br />
∠2 = __________<br />
12.<br />
1<br />
2<br />
3<br />
A. If ∠1 = 37° and ∠2 = 45° then ∠3 = __________<br />
B. if ∠1 + ∠3 = 108°, then ∠2 = __________
13.<br />
<strong>Angles</strong> page #11<br />
2<br />
1<br />
A. Write a formula (equation) for ∠1 and ∠2: __________________________<br />
B. If ∠1 = 3x+6 and ∠2 = 4x, then:<br />
x = __________<br />
∠1 = __________<br />
∠2 = __________<br />
C. If ∠1 = 2x-4 and ∠2 = x-2, then:<br />
x = __________<br />
∠1 = __________<br />
∠2 = __________<br />
14. Find the measure of each numbered angle.<br />
48°<br />
1<br />
101° 2<br />
3<br />
104°<br />
4 5<br />
∠1 = __________ ∠2 = __________ ∠3 = __________<br />
∠4 = __________<br />
∠5 = __________
<strong>Angles</strong> page #12<br />
Vertical <strong>Angles</strong><br />
∠3 and ∠4 are a pair of vertical angles.<br />
They have no common side.<br />
3 4<br />
They are formed by two intersecting lines.<br />
Vertical angles are equal.<br />
Problems:<br />
1.<br />
Z<br />
V<br />
1<br />
U<br />
5<br />
4<br />
2<br />
3<br />
Y<br />
W<br />
X<br />
A. ∠2 and ∠ __________ form a vertical pair.<br />
B. Do ∠1 and ∠3 form a vertical pair? ________<br />
C. Do ∠1 and ∠4 form a vertical pair? ________<br />
D. What angle forms a vertical pair with ∠1? _______<br />
2. Find the measures of angles 1 through 7.<br />
4<br />
1<br />
106°<br />
79°<br />
42°<br />
3<br />
2<br />
59°<br />
7<br />
67°<br />
41°<br />
6<br />
5<br />
∠1 = _______ ∠2 = _______ ∠5 = _______<br />
∠3 = _______<br />
∠6 = _______<br />
∠4 = _______<br />
∠7 = _______
3. Find x and the measure of each angle:<br />
A.<br />
<strong>Angles</strong> page #13<br />
8x<br />
10x-30<br />
B.<br />
x+45<br />
2x<br />
C.<br />
6x<br />
10x-32<br />
D.<br />
3x<br />
x+18
4.<br />
<strong>Angles</strong> page #14<br />
D<br />
A<br />
4<br />
3<br />
1<br />
B<br />
2<br />
C<br />
A. Name two pairs of vertical angles:<br />
__________ and __________<br />
E<br />
__________ and __________<br />
B. The angle vertical to ∠DBE is __________.<br />
C. Name 4 pairs of supplementary angles:<br />
__________ and __________<br />
__________ and __________<br />
__________ and __________<br />
__________ and __________<br />
D. If ∠CBE = 42°, then: ∠ABD = __________<br />
∠ABC = __________<br />
∠DBC = __________<br />
5. Find x and the measure of each angle.<br />
A. B. C.<br />
3x+20 5x 5x-18 90-4x<br />
4x<br />
5x-27
Proving that Vertical <strong>Angles</strong> are Equal<br />
A theorem is a statement that you can prove.<br />
Theorem: Vertical <strong>Angles</strong> are equal.<br />
<strong>Angles</strong> page #15<br />
1<br />
3<br />
4<br />
2<br />
Prove:<br />
∠1 = ∠2 and ∠ 3 = ∠4
<strong>Angles</strong> page #16<br />
Review complementary and supplementary Word Problems<br />
1. Two angles are complementary and one angle is 4 times as large as the other angle.<br />
Find the number of degrees in each angle.<br />
2. The measure of an angle is 4 more than the measure of its supplement. Find the<br />
measure of each angle.<br />
3. The measure of an angle is 8 times the measure of its complement. Find the<br />
measure of each angle.<br />
4. The measure of an angle is 8 more than the measure of its complement. Find the<br />
measure of each angle.<br />
5. Two angles are supplementary. One angle is twice the other. find the number of<br />
degrees in each angle.