Angles of Triangles Objectives Find angle measures in triangles Key Vocabulary Supplementary Angles Complementary Angles Exterior angles Interior angles Additional Vocabulary Theorem Mathematics a theoretical proposition statement or formula embodying something to be proved from other propos ID: 765750
Download Presentation The PPT/PDF document "Angles of Triangles" is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
Angles of Triangles
Objectives Find angle measures in triangles.
Key Vocabulary Supplementary Angles Complementary Angles Exterior angles Interior angles
Additional Vocabulary Theorem - Mathematics. a theoretical proposition, statement, or formula embodying something to be proved from other propositions or formulas. 2. a rule or law, especially one expressed by an equation or formula . (Dictionary.com)
Theorems 4.1 Triangle Sum Theorem 4.2 Exterior Angle Theorem
Measures of Angles of a Triangle The word “triangle” means “three angles” When the sides of a triangles are extended, however, other angles are formed The original 3 angles of the triangle are the interior angles The angles that are adjacent to interior angles are the exterior angles Each vertex has a pair of exterior angles Original Triangle Extend sides Interior Angle Exterior Angle Exterior Angle
Triangle Interior and Exterior Angles Smiley faces are interior angles and hearts represent the exterior angles Each vertex has a pair of congruent exterior angles; however it is common to show only one exterior angle at each vertex.
Triangle Interior and Exterior Angles )) ))) ( A B C ( )) (( D E F Interior Angles Exterior Angles (formed by extending the sides)
Triangle Sum Theorem The Triangle Angle-Sum Theorem gives the relationship among the interior angle measures of any triangle.
Triangle Sum Theorem If you tear off two corners of a triangle and place them next to the third corner, the three angles seem to form a straight line. You can also show this in a drawing.
Draw a triangle and extend one side. Then draw a line parallel to the extended side, as shown. The three angles in the triangle can be arranged to form a straight line or 180°. Two sides of the triangle are transversals to the parallel lines. Triangle Sum Theorem
Triangle Sum Theorem The sum of the measures of the angles of a triangle is 180 °. m X + m Y + mZ = 180 ° X Y Z
Triangle Sum Theorem
Given m A = 43 ° and m B = 85 °, find mC. ANSWER C has a measure of 52°. CHECK Check your solution by substituting 52 ° for m C. 43° + 85 ° + 52° = 180° SOLUTION m A + m B + m C = 180 ° Triangle Sum Theorem 43 ° + 85 ° + m C = 180 ° Substitute 43 ° for m A and 85 ° for m B . 128 ° + m C = 180 ° Simplify. m C = 52 ° Simplify. 128 ° + m C – 128 ° = 180 ° – 128 ° Subtract 128 ° from each side. Example 1
A. Find p in the acute triangle. 73° + 44° + p ° = 180° 117 + p = 180 p = 63 –117 –117 Triangle Sum Theorem Subtract 117 from both sides. Example 2a
B. Find m in the obtuse triangle. 23° + 62° + m ° = 180° 85 + m = 180 m = 95 –85 –85 Triangle Sum Theorem Subtract 85 from both sides. 23 62 m Example 2b
A. Find a in the acute triangle. 88° + 38° + a ° = 180° 126 + a = 180 a = 54 –126 –126 88° 38° a ° Triangle Sum Theorem Subtract 126 from both sides. Your Turn:
B. Find c in the obtuse triangle. 24° + 38° + c ° = 180° 62 + c = 180 c = 118 –62 –62 c ° 24° 38° Triangle Sum Theorem. Subtract 62 from both sides. Your Turn:
2 x ° + 3 x ° + 5 x° = 180° 10 x = 180 x = 18 10 10 Find the angle measures in the scalene triangle. Triangle Sum Theorem Simplify. Divide both sides by 10. The angle labeled 2 x ° measures 2(18°) = 36°, the angle labeled 3 x° measures 3(18°) = 54°, and the angle labeled 5 x° measures 5(18°) = 90°.Example 3
3 x ° + 7 x ° + 10 x° = 180° 20 x = 180 x = 9 20 20 Find the angle measures in the scalene triangle. Triangle Sum Theorem Simplify. Divide both sides by 20. 3 x ° 7 x ° 10 x° The angle labeled 3 x ° measures 3(9°) = 27°, the angle labeled 7 x ° measures 7(9°) = 63°, and the angle labeled 10 x ° measures 10(9°) = 90°. Your Turn:
Find the missing angle measures. Find first because the measure of two angles of the triangle are known. Angle Sum Theorem Simplify. Subtract 117 from each side. Example 4:
Answer: Angle Sum Theorem Simplify. Subtract 142 from each side. Example 4:
Find the missing angle measures. Answer: Your Turn:
m DAB + 35 ° = 90 ° Substitute 35 ° for m ABD . m DAB = 55 ° Simplify. m DAB + 35° – 35 ° = 90° – 35° Subtract 35 ° from each side. ∆ ABC and ∆ ABD are right triangles. Suppose m ABD = 35°. Find m DAB . a. b. Find m BCD . 55 ° + m BCD = 90 ° Substitute 55 ° for m DAB . m BCD = 35 ° Subtract 55 ° from each side. SOLUTION m DAB + m ABD = 90 ° a. m DAB + m BCD = 90 ° b. Example 5
ANSWER 65 ° ANSWER 75 ° ANSWER 50 ° Find m A . 1. Find m B . 2. Find m C . 3. Your Turn:
Corollary 4.1 Substitution Subtract 20 from each side. Answer: GARDENING The flower bed shown is in the shape of a right triangle. Find if is 20 . Example 6:
Answer: The piece of quilt fabric is in the shape of a right triangle. Find if is 62 . Your Turn:
Investigating Exterior Angles of a Triangles B A A B C You can put the two torn angles together to exactly cover one of the exterior angles
Exterior Angles and Triangles An exterior angle is formed by one side of a triangle and the extension of another side (Example 1 ). The interior angles of the triangle not adjacent to a given exterior angle are called the remote interior angles (i.e. 2 and 3). 1 2 3 4
Exterior Angle Theorem The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles. m 1 = m 2 + m 3 1 2 3 4
ANSWER 1 has a measure of 130 ° . SOLUTION m 1 = m A + m C Exterior Angle Theorem Given m A = 58 ° and mC = 72°, find m1. Substitute 58 ° for m A and 72 ° for m C . = 58 ° + 72 ° Simplify. = 130 ° Example 7
ANSWER 120 ° ANSWER 155 ° ANSWER 113 ° Find m 2 . 1. Find m 3 . 2. Find m 4 . 3. Your Turn:
Find the measure of each numbered angle in the figure. Exterior Angle Theorem Simplify. Substitution Subtract 70 from each side. If 2 s form a linear pair, they are supplementary. Example 8:
Exterior Angle Theorem Subtract 64 from each side. Substitution Subtract 78 from each side. If 2 s form a linear pair, they are supplementary. Substitution Simplify. Example 8: m ∠ 1=70 m ∠ 2=110
Subtract 143 from each side. Angle Sum Theorem Substitution Simplify. Answer: Example 8: m ∠ 1=70 m ∠ 2=110 m ∠ 3=46 m ∠ 4=102
Find the measure of each numbered angle in the figure. Answer: Your Turn:
Success Starter Solve for x :
Unit 6 Parallel Lines Learn about parallel line relationships Prove lines parallel Describe angle relationship in polygons
Parallel Lines Coplanar lines that do not intersect. m n Skew lines are non-coplanar, non-intersecting lines. m || n p q
The Transversal Any line that intersects two or more coplanar lines. r s t
Lecture 2 Objectives Learn the special angle relationships …when lines are parallel
When parallel lines are cut by a transversal… Corresponding ’s 1 5 Alternate Interior ’s 4 5Same Side Interior ’s Suppl. 4 suppl. 6 r s t 1 2 3 4 5 6 7 8
If two parallel lines are cut by a transversal, then corresponding angles are congruent. r s t 1 2 3 4 5 6 7 8
If two parallel lines are cut by a transversal, then alternate interior angles are congruent. r s t 1 2 3 4 5 6 7 8
If two parallel lines are cut by a transversal, then same side interior angles are supplementary. r s t 1 2 3 4 5 6 7 8
A line perpendicular to one of two parallel lines is perpendicular to the other. r s t
If two lines are cut by a transversal so that corresponding angles are congruent, then the lines are parallel. 1 2 m n If 1 2, then m || n.
If two lines are cut by a transversal so that alternate interior angles are congruent, then the lines are parallel. 1 2 m n If 1 2, then m || n.
If two lines are cut by a transversal so that same side interior angles are supplementary, then the lines are parallel. 1 2 m n If 1 suppl 2, then m || n.
In a plane, two lines perpendicular to the same line are parallel. m n If t m and t n , then m || n. t
Two lines parallel to the same line are parallel to each other m n p If p m and m n, then p n
Ways to Prove Lines are Parallel Corresponding angles are congruent Alternate interior angles are congruent Same side interior angles are supplementary In a plane, that two lines are perpendicular to the same line Both lines are parallel to a third line