Day 19 – Vertical and interior angles

Introduction When two or more lines intersect at a given point, they form angles at the intersection point. These angles possess some properties in reference to each other. In this lesson we are going to discover the properties of such angles together with their algebraic interpretations.

Vocabulary Transversal A line that intersects two or more lines, especially parallel lines, at two more distinct points. Supplementary angles A pair of angles whose sum is 180°

Vertical angles These are two angles formed when two lines intersect, and they are opposite each other. They are congruent; they have equal measures. In the figure above, ∠𝑝 𝑎𝑛𝑑 ∠𝑞 are vertical angles. Similarly, ∠𝑟 𝑎𝑛𝑑 ∠𝑠 are vertical angles. Note that ∠𝒑=∠𝒒 and ∠𝒓=∠𝒔 p q r s

Example 1 Find the size of the angle marked z Example 1 Find the size of the angle marked z. Solution 𝑧 and 78° are vertical angles. Vertical angles are equal, therefore 𝑧=78°. 𝑧 78°

Example 2 Find all the unknown angles formed in the figure below when the lines intersect as shown. Solution x and 49° are vertical angles, therefore 𝑥=49°. 𝑥 49° y z

𝑦 and 49° are angles on a straight line 𝑦 and 49° are angles on a straight line. Recall that angles on a straight line add up to 180°, therefore 𝑦+49°=180° 𝑦=131° y and z are vertical angles, vertical angles are congruent, hence 𝑧=131°

Example 3 In the figure below, find the value of t and hence or otherwise find angles ∠FOI and ∠FOG . 2𝑡−51 ° 𝑡 +49 ° F G H I O 31°

Solution ∠FOI and ∠FOG are vertical angles and therefore equal Solution ∠FOI and ∠FOG are vertical angles and therefore equal. ∠HOI=∠FOG 2𝑡−51= 𝑡 +49 𝒕=𝟏𝟎𝟎° ∠FOG= 2𝑡−51 but 𝑡=100°. Therefore we substitute the value of t in the equation. ∠𝐅𝐎𝐆= 2(100)−51=𝟏𝟒𝟗° ∠FOI and ∠GOH are vertical angles, therefore they are equal. ∠𝐅𝐎𝐈=𝟑𝟏°

Interior angles They are formed when a transversal intersects a pair of parallel lines and they are found between the parallel lines. In the figure above JK∥LM. 1, 2, 3 and 4 form interior angles. 1 2 4 3 J K L M

Consecutive interior angles In the figure above JK∥LM 1 and 3 form a pair of consecutive interior angles. Similarly 2 and 4 form another pair of consecutive interior angles. They are just interior angles located on the same side of the transversal intersecting a pair of parallel lines. Consecutive interior angles are supplementary. 1 2 4 3 J K L M

Example 4 Given that AB∥CD, find the size of the angle marked b Example 4 Given that AB∥CD, find the size of the angle marked b. Solution 146° and b are consecutive interior angles and therefore supplementary. 146°+𝑏=180° b 146° A B C D

𝒃=180°−146°=𝟑𝟒° Example 5 Given that RS∥TU, find the value of x. 2𝑥+4 ° 4𝑥−10 ° R S T U

Solution 2𝑥+4 ° and 4𝑥−10 ° are consecutive interior angles, therefore: 2𝑥+4 °+ 4𝑥−10 °=180° 6𝑥−6=180° 6𝑥=186° 𝒙=𝟑𝟏°

homework If PQ∥RS and KL∥MN . Find the angle marked x and y using the concept of vertical and interior angles. x 111° P Q R S y

Answers to homework 𝑥=69°, 𝑦=111°

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