13
Adjacent and vertical angles and their properties
Textbook: pp. 34–35
GoalKnow and apply the properties of adjacent and vertical angles.
New words
adjacent angles · qo‘shni burchaklarvertical angles · vertikal burchaklarangle between lines · to‘g‘ri chiziqlar orasidagi burchakproperty · xossa
Explanation
If two angles share one side and their other two sides together make up one line, they are called adjacent angles. Property: the sum of adjacent angles is 180°. If the sides of one angle are the extensions of the sides of another, the angles are called vertical; two intersecting lines form two pairs of vertical angles. Property: vertical angles are equal. Proof: let α and β be vertical and γ adjacent to both; α + γ = 180° and γ + β = 180°, so α = β. Of the angles formed by two intersecting lines, the one not greater than 90° is called the angle between the lines.
Worked examples
If one of two adjacent angles is 65°, the other is 180° − 65° = 115°.
Two lines intersect and one angle is 30° bigger than its neighbour: x + (x + 30°) = 180°, so x = 75° and 105°. The four angles are 75°, 105°, 75°, 105°.
Class activity
“Crossing pencils”: cross two pencils and measure the four angles with a protractor: which are equal? Which pairs add to 180°?
Practice
1
One of two adjacent angles is 112°. Find the other (degrees).
68
2
Two lines intersect and one angle is 54°. Find the other three (starting with the smaller).
54°, 126°, 126°
3
Adjacent angles are in the ratio 2 : 7. How many degrees is the smaller one?
40
4
Why are vertical angles equal?
Both are adjacent to the same angle, so each equals 180° minus that angle.