D AY 16 – C OMPLEMENTARY AND SUPPLEMENTARY ANGLES
I NTRODUCTION In geometry, we will come across pairs of angles whose sum is either 90° or 180° . In this lesson, we are going to take a look at such pairs of angles and some of their algebraic properties.
V OCABULARY Angle The amount of turn between two rays having a common endpoint.
Complementary angles These are two angles whose sum is 90° . For example, 30° 𝑏𝑜𝑒 60°; 15° 𝑏𝑜𝑒 75°, 22.5° 𝑏𝑜𝑒 67.5°. It is clear that they add up to 90°. 60° + 30° = 90°; 15° + 75° = 90°; 22.5° + 67.5° = 90° 𝑦 y Angle x is called the complement of angle y. Similarly, angle y is the complement of angle x .
Example 1 Find the complement of the following angles: (a) 70° (b) 51° (c) 13.5° Solution Complementary angles add up to 90° (a) 90° − 70° = 20° (b) 90° − 51° = 39° (c) 90° − 13.5° = 76.5° Each angle is the complement of the other angle.
Example 2 Given that ∠𝑏 and ∠𝑐 are complementary such that ∠𝑏 = 3𝑦 − 3 and ∠𝑐 = 4𝑦 + 2 . Find the measure of each angle. Solution Complementary angles add up to 90° . Therefore the sum of the two angles should be equal to 90°. 3𝑦 − 3 + 4𝑦 + 2 = 90° Simplifying the equation, we have: 3𝑦 + 4𝑦 − 1 = 90 Solving for x , we have: 7𝑦 = 91 𝑦 = 13
In order to obtain the angle measure of each angle, we substitute the value of x in the angle measures. ∠𝑏 = 3 13 − 3 = 𝟒𝟕° and ∠𝑐 = 4 13 + 2 = 𝟔𝟓°
Example 3 Find the value of x in the diagram below. 2𝑦 − 3 4𝑦 + 3 Solution The angles are complementary, therefore, we have: 2𝑦 − 3 + 4𝑦 + 3 = 90° 6𝑦 = 90° 𝑦 = 15°
Supplementary angles These are two angles whose sum is 180° . For example, 30° 𝑏𝑜𝑒 150°; 15° 𝑏𝑜𝑒 165°, 22.5° 𝑏𝑜𝑒 157.5°. It is clear that they add up to 90°. 30° + 150° = 180°; 15° + 165° = 180°; 22.5° + 157.5° = 90° 𝑦 y Angle x is called the supplement of angle y. Similarly, angle y is the supplement of angle x .
Example 4 Find the supplement of the following angles: (a) 10° (b) 111° (c) 43.5° Solution Supplementary angles add up to 180° (a) 180° − 10° = 170° (b) 180° − 111° = 69° (c) 180° − 43.5° = 136.5° Each angle is the supplement of the other angle. Two supplementary angles form a linear pair .
Example 5 Given that ∠𝑛 and ∠𝑜 are supplementary such that ∠𝑛 = 2𝑦 − 3 and ∠𝑜 = 3𝑦 + 8 . Find the measure of each angle. Solution Supplementary angles add up to 180° . Therefore the sum of the two angles should be equal to 180°. 2𝑦 − 3 + 3𝑦 + 8 = 180° Simplifying the equation, we have: 2𝑦 + 3𝑦 + 5 = 180 Solving for x , we have: 5𝑦 = 175 𝑦 = 35
In order to obtain the angle measure of each angle, we substitute the value of x in the angle measures. ∠𝑛 = 2 35 − 3 = 𝟕𝟖° and ∠𝑜 = 3 35 + 8 = 𝟐𝟐𝟒°
Example 6 Find the value of x in the diagram below. 3𝑦 + 20 5𝑦 − 4 Solution The angles are supplementary, therefore, we have: 3𝑦 + 20 + 5𝑦 − 4 = 180° 8𝑦 + 16 = 180° 𝑦 = 20.5°
Example 7 Given that ∠𝑌 and ∠𝑍 are supplementary and the measure of ∠𝑌 is five times that of ∠𝑍 . Find the measure of both angles. Solution Since they are supplementary, their sum must be 180° and the measure of ∠𝑌 is five times that of ∠𝑍 . ∠𝑌 + ∠𝑍 = 180° 5∠𝑍 + ∠𝑍 = 180° 6∠𝑍 = 180° ∠𝑍 = 𝟒𝟏° This means that ∠𝑌 = 5 × 30° = 𝟐𝟔𝟏°
Angles on a straight line Angles on a straight line always add up to 180° . In the diagram below, RS is a straight line. 𝑐 c 𝑏 𝑇 𝑆 ∠𝑏 + ∠𝑐 + ∠𝑑 = 180°
Example 8 Find the measure of angle 𝛾 in the figure below, given that AB is a straight line. 𝛾 65° A B Solution Angles on a straight line add up to 180°, therefore we have: 𝛾 + 65° = 180° 𝛾 = 180° − 65° = 115°
H OMEWORK Consider the figure below. DG and EH are straight lines. ∠HOG = 53° and ∠EOG = 37° . H 𝑡 53° D G 𝑠 O 37° E F
(a) Find the angle measure represented by r and s. (b) Identify the complement of ∠DO E. (c) Identify two supplements of ∠DO H
A NSWERS TO HOMEWORK (a) ∠𝑠 = 53°, ∠𝑡 = 127° (b) ∠EO F (c) ∠DOE and ∠HOG
THE END
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