Introduction
When a transversal line crosses two or more parallel lines, it creates special pairs of angles. Two important types are alternate angles and corresponding angles.
In this lesson, we will learn how to identify these angles, draw diagrams showing them, and calculate their values using angle properties.
Alternate Angles
Definition
Alternate angles are pairs of angles that are on opposite sides of the transversal and between the two parallel lines. They are equal in measure when the lines are parallel.
Properties
Key properties of alternate angles:
Equal Measure
Alternate angles are equal when lines are parallel:
\[ \text{Alternate angle } a = \text{Alternate angle } b \]Z-Shape
They form a 'Z' shape between the parallel lines.
Calculating Values
To find the value of alternate angles:
- Identify the parallel lines and transversal
- Locate the 'Z' shape formed by the angles
- Set the alternate angles equal to each other
- Solve for the unknown angle
Example: If one alternate angle is 65°, what is its pair?
Since alternate angles are equal:
\[ \text{Both angles} = 65° \]Corresponding Angles
Definition
Corresponding angles are pairs of angles that are in matching corners (same position) at each intersection where the transversal crosses the parallel lines. They are equal in measure when the lines are parallel.
Properties
Key properties of corresponding angles:
Equal Measure
Corresponding angles are equal when lines are parallel:
\[ \text{Corresponding angle } a = \text{Corresponding angle } b \]F-Shape
They form an 'F' shape along the parallel lines.
Calculating Values
To find the value of corresponding angles:
- Identify the parallel lines and transversal
- Locate the 'F' shape formed by the angles
- Set the corresponding angles equal to each other
- Solve for the unknown angle
Example: If one corresponding angle is 112°, what is its pair?
Since corresponding angles are equal:
\[ \text{Both angles} = 112° \]Practice Exercise
Question 1
In the diagram below, lines AB and CD are parallel. Find the values of angles x and y.
Solution:
Angle x is alternate to the 75° angle, so:
\[ x = 75° \]Angle y is corresponding to the 75° angle, so:
\[ y = 75° \]Question 2
In the diagram below, lines PQ and RS are parallel. Calculate the values of angles a, b, and c.
Solution:
Angle a is corresponding to the 110° angle, so:
\[ a = 110° \]Angle b is alternate to angle a, so:
\[ b = 110° \]Angle c forms a straight line with angle b, so:
\[ c = 180° - 110° = 70° \]Question 3
Draw a diagram showing two parallel lines crossed by a transversal, labeling one pair of alternate angles (both 60°) and one pair of corresponding angles (both 120°).
Your diagram should show:
- Two parallel lines crossed by a transversal
- One pair of alternate angles marked as 60°
- One pair of corresponding angles marked as 120°
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