krit.club logo

Geometry and Measurement - Properties of Parallel and Perpendicular Lines

Grade 7IB

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

•

Parallel Lines (L1∣∣L2L_1 || L_2): These are lines in the same plane that never meet, no matter how far they are extended. Visually, they resemble railroad tracks or the opposite edges of a ruler, maintaining a constant distance from each other at all points.

•

Perpendicular Lines (L1⊥L2L_1 \perp L_2): Lines that intersect at a perfect right angle (90∘90^\circ). Visually, this creates a 'T-shape' or a '+' shape, forming a perfect square corner at the point of intersection.

•

Transversal Line: A line that crosses at least two other lines. When a transversal intersects parallel lines, it creates eight specific angles that share predictable relationships based on their positions.

•

Corresponding Angles: These angles are in the same relative position at each intersection where a transversal crosses parallel lines. If the lines are parallel, these angles are equal. Visually, they form an 'F' shape (which can be backwards or upside down) along the lines.

•

Alternate Interior Angles: These angles are located between the two parallel lines but on opposite sides of the transversal. If the lines are parallel, these angles are equal. Visually, they can be identified by looking for a 'Z' or 'N' shape.

•

Co-interior (Consecutive Interior) Angles: These are pairs of angles located between the parallel lines and on the same side of the transversal. Unlike corresponding or alternate angles, these are not equal; instead, they are supplementary, meaning they add up to 180∘180^\circ. Visually, they are nestled within a 'C' or 'U' shape.

•

Vertically Opposite Angles: When two lines intersect, the angles opposite each other at the vertex are always equal. Visually, this creates an 'X' shape where the top angle equals the bottom, and the left equals the right.

•

Linear Pair: When two angles are adjacent and sit on a straight line, they form a linear pair. These angles are supplementary and their measures always sum to 180∘180^\circ.

📐Formulae

If L1∣∣L2L_1 || L_2, then ∠Corresponding1=∠Corresponding2\angle \text{Corresponding}_1 = \angle \text{Corresponding}_2

If L1∣∣L2L_1 || L_2, then ∠Alternate Interior1=∠Alternate Interior2\angle \text{Alternate Interior}_1 = \angle \text{Alternate Interior}_2

If L1∣∣L2L_1 || L_2, then ∠Co-interior1+∠Co-interior2=180∘\angle \text{Co-interior}_1 + \angle \text{Co-interior}_2 = 180^\circ

For L1⊥L2L_1 \perp L_2, the angle of intersection is 90∘90^\circ

∠Linear Pair1+∠Linear Pair2=180∘\angle \text{Linear Pair}_1 + \angle \text{Linear Pair}_2 = 180^\circ

∠Vertically Opposite1=∠Vertically Opposite2\angle \text{Vertically Opposite}_1 = \angle \text{Vertically Opposite}_2

💡Examples

Problem 1:

In the diagram, line ABAB is parallel to line CDCD. A transversal line EFEF intersects ABAB at point PP and CDCD at point QQ. If ∠APQ=(2x+10)∘\angle APQ = (2x + 10)^\circ and ∠DQP=(3x−20)∘\angle DQP = (3x - 20)^\circ are alternate interior angles, find the value of xx and the measure of each angle.

Solution:

  1. Since AB∣∣CDAB || CD, alternate interior angles are equal: (2x+10)=(3x−20)(2x + 10) = (3x - 20).
  2. Subtract 2x2x from both sides: 10=x−2010 = x - 20.
  3. Add 2020 to both sides: x=30x = 30.
  4. Substitute xx back into the expressions: ∠APQ=2(30)+10=70∘\angle APQ = 2(30) + 10 = 70^\circ and ∠DQP=3(30)−20=70∘\angle DQP = 3(30) - 20 = 70^\circ.

Explanation:

Because the lines are parallel, we can use the property that alternate interior angles (the 'Z' shape) are equal in measure to set up an algebraic equation and solve for the unknown.

Problem 2:

Given two parallel lines intersected by a transversal, two co-interior angles are represented by xx and 4x4x. Find the measure of the larger angle.

Solution:

  1. Co-interior angles between parallel lines are supplementary: x+4x=180∘x + 4x = 180^\circ.
  2. Combine like terms: 5x=180∘5x = 180^\circ.
  3. Divide by 55: x=1805=36∘x = \frac{180}{5} = 36^\circ.
  4. Find the larger angle: 4x=4×36=144∘4x = 4 \times 36 = 144^\circ.

Explanation:

We use the co-interior angle property (the 'C' shape), which states that these angles sum to 180∘180^\circ, to create a linear equation and solve for the variable.