Lesson 1: Angles & Polygons

LO: Identify, calculate, and use angle properties to solve problems involving triangles, quadrilaterals, and polygons

1. Types of Angles ( Basics, Parts & Properties )

• An angle measures a turn using degrees.

• It is made up of two rays ( straight lines) that meet at the vertex.

Finding Missing Angles

1. Subtract the known angle or the sum of known angles from 180 degrees if on a straight line, or 360 degrees if around a point.

E.g. Calculation for missing angles on a straight line

A straight line is split into two angles

• One is 57 degrees

• The other angle is missing, labelled as “a”

1. $180-57=123$ degrees

2. $a=123$ degrees

E.g. Calculation for missing angles around a point

Imagine 4 angles meeting at a point

• Three of the known angles are known: 115 degrees, 78 degrees, and 34 degrees

• The 4th angle is missing, labelled as “x”

1. $115+78+34=227$ degrees

2. $360-227=133$ degrees

3. $x=133$ degrees

2. Vertically Opposite Angles

• Vertically opposite angles are identified when two straight lines cross forming two pairs of angles, vertically opposite of each other and equal in measure.

Finding Vertically Opposite Angles

NOTE: ALL ANGLES MUST SUM UP TO 360 DEGREES

First Method

1. If you know one angle, the angle vertically opposite of it, is identical

2. Subtract the given angle from 180 degrees because it is on a straight line

3. The angle opposite of it will be identical

E.g. Calculation for finding vertically opposite angles ( 1st method )

Vertically opposite angles make an “x'“ shape

• The top angle is 64 degrees

• The bottom angle is unknown and labelled as “a”

• The left angle is unknown and labelled as “b”

• The right angle is unknown and labelled as “c”

1. Angle “a” is vertically opposite of the top angle, therefore, $a=64$ degrees

2. Subtract 64 from 180, $180-64=116$ . Angle “b” is 116 degrees

3. And angle “c” is vertically opposite of angle “b”, therefore angle “c” is 116 degrees

Second Method

1. If you known one angle, the angle vertically opposite of it, is identical

2. Add both of them together

3. Subtract the sum from 360 degrees

4. Half it because you are finding what two angles of the same measure are

E.g. Calculation for finding vertically opposite angles ( 2nd method )

1. Angle “a” is vertically opposite of the top angle, therefore, $a=64$ degrees

2. Double 64, $64+64=128$

3. Subtract 128 from 360 degrees, $360-128=232$

4. Divide 232 by 2, $\frac{232}{2}=116$degrees . Angle “b’ and “c” are both 116 degrees each as they are vertically opposite of each other

3. Types & Properties of Triangles and Quadrilaterals

Triangles

• A triangle is a 3-sided polygon whose interior angles sum up to 180 degrees.

Finding Missing Angles in Triangles

NOTE: ALL ANGLES MUST SUM UP TO 180 DEGREES, NO MATTER WHAT TYPE IT IS

Equilateral Triangle

Method

1. Divide 180 by 3, because all 3 sides & interior angles are equal, giving you a quotient of 60 degrees for each side

Isosceles Triangle

First Method ( If top angle is known )

1. Subtract the known angle from 180 degrees

2. Divide the result by 2 to know what the two base angles are individually

E.g. Calculation for finding missing angles in triangles ( 1st method )

• Top angle is 52 degrees

1. Because the top angle is 52 degrees, subtract 52 from 180 degrees, $180-52=128$

2. Divide 128 by 2, to find the two base angles which are both equal, $\frac{128}{2}=64$ . The two base angles are each 64 degrees

Second Method ( If one base angle is known )

1. Double the known base angle ( since they are equivalent )

2. Subtract the answer from 180 degrees to find the top angle

E.g. Calculation for finding missing angles in triangles ( 2nd method )

• The base angle is 23 degrees

1. Because the two base angles are the same, double it, $23+23=46$

2. Subtract 46 from 180 degrees, to find the top angle, $180-46=134$ . The top angle is 134 degrees

Scalene Triangle

Method

1. Add the two known angles together

2. Subtract the sum of the two known angles from 180, to get the missing angle

E.g. Calculation for finding missing angles in triangles

• The two known angles are 45 & 89 degrees

1. Add 45 and 89 degrees together, $45+89=134$

2. Subtract 134 from 180 degrees to find the missing angle, $180-134=46$ . The missing angle is 46 degrees

Right Triangle

NOTE: ONE ANGLE IS A RIGHT ANGLE, HENCE WHY IT IS CALLED A RIGHT TRIANGLE

Method

1. Add the right angle with the other known angle

2. Subtract the sum from 180 degrees

E.g. Calculation for finding missing angles in triangles

• The other known angle is 67 degrees

1. Add 90 and 67 degrees together, $90+67=157$

2. Subtract 157 from 180 degrees to find the missing angle, $180-157=23$ . The missing angle is 23 degrees

Acute Triangle

NOTE: ALL THREE ANGLES ARE ACUTE ANGLES

Method

1. Add the two known angles together

2. Subtract the sum from 180 degrees

E.g. Calculation for finding missing angles in triangles

• The two known acute angles are 89 & 45 degrees

1. Add 89 and 45 degrees together, $89+45=134$

2. Subtract 134 from 180 degrees to find the missing angle, $180-134=46$ . The missing angle is 46 degrees

Obtuse Triangle

NOTE: ONE ANGLE IS OBTUSE, THE OtHER TWO ARE ACUTE

Method

1. Add the two known angles together

2. Subtract the sum from 180 degrees

E.g. Calculation for finding missing angles in triangles

• The two known angles are 123 and 23 degrees

1. Add 123 and 23 degrees together, $123+23=146$

2. Subtract 146 from 180 degrees to find the missing angle, $180-146=34$ . The missing angle is 34 degrees

Quadrilaterals

• A quadrilateral is a four-sided polygon whose interior angles sum up to 360 degrees.

Calculating Missing Angles in Quadrilaterals

NOTE: ALL ANGLES MUST SUM UP TO 360 DEGREES

Squares & Rectangles

Method

1. Divide 360 by 4, because all 4 sides and angles are equal, which gives you a quotient of 90 degrees, for each side

Irregular Quadrilaterals

1. Add the 3 known angles

2. Subtract the sum from 360 degrees

E.g. Calculation for finding missing angles in quadrilaterals

• The 3 known angles are 152, 83, and 41 degrees

1. Add 152, 83, and 41 degrees together, $152+83+41=276$

2. Subtract 276 from 360 degrees to find the missing angle, $360-276=84$ . The missing angle is 84 degrees

4. Types & Properties of Polygons

• A polygon is a 2D shape made of straight lines.

Sum of Interior Angles of Polygons

Formula

Sum of Angles = ( n - 2 ) x 180 degrees, n is the number of sides

E.g. Calculation for finding the sum of interior angles in hexagons

Finding Individual Angle of a Polygon

Formula

Interior Angle = Sum divided by n