2.2AnglesFormedbyParallelLines.notebook
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Section 2.2 Angles Formed by Parallel Lines
Goal: Prove properties of angles formed by parallel lines and a transversal, and use these properties to solve problems.
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Recall: Corresponding angles created by a transversal and parallel lines are EQUAL.
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Interior Angles on the Same-Side of the Transversal
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Exterior Angles on the Same-Side of the Transversal
Alternate Exterior Angles
Alternate Interior Angles
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Before we get into investigating the angle relationships created with parallel lines, we need to investigate how to CONSTRUCT parallel lines. How could we do this using a compass and straightedge?
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To CONSTRUCT parallel lines using a straight edge and a compass:
1. Draw a line and a point P NOT on the line.
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2. Draw a line through P intersecting the line at Q.
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3. Using a compass, construct an arc centered at Q and passing through both lines. Label the intersection points R & S.
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4. Draw another arc, centered at P, with the same radius as the arc QR. Label the intersection point T.
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5. Draw a third arc, with center T, and radius RS that intersects the arc you drew in step 4. Label the point of intersection W.
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6. Draw the line that passes through P and W. Show that PW||QS.
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Why does this technique ensure that the two lines you drew are parallel?
Your Turn: Follow the 6 steps above and CONSTRUCT parallel lines of your own.
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Textbook page 75
Make conjectures that involve the interior angles formed by parallel lines and a transversal. Prove your conjecture.
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Conjecture 1: When a transversal intersects a pair of parallel lines, the alternate interior angles are equal. Use a two column proof to prove the conjecture.
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Conjecture 2: When a transversal intersects a pair of parallel lines, the interior angles on the same side of the transversal are supplementary. Use a two column proof to prove the conjecture.
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Justification
Statement
Corresponding Angles Angles formed by straight line are supplementary.
Substitution
Vertically Opposite Angles Substitution
The conjecture is proven.
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Conjecture 3: Alternate exterior angles are equal. Statement
Justification
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Conjecture 3: Alternate exterior angles are equal. Statement
Justification Corresponding Angles
Vertically Opposite Angles
Transitive Property
The conjecture is proven.
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2.2AnglesFormedbyParallelLines.notebook
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RECAP: Properties of Pairs of Angles Formed by Cutting Parallel Lines with a Transversal.
NOTE: If any of these properties hold true then the two lines intersected with a transversal must be parallel. All these properties can now be used in order to prove if lines are parallel.
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Using Reasoning to Determine Unknown Angles Determine the measures of a, b, c, and d.
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Justification
Statement
Vertically opposite angles are equal.
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Justification
Statement
c and a are interior angles on the same side of the transversal. Since the lines are parallel, c and a are supplementary.
c and d are alternate interior angles. Since the lines are parallel, c and d are equal.
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What would be a different strategy to use to determine the measures of angles b and d?
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Example 3:
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(Textbook page 77)
One side of a cellphone tower will be built as shown. Use the angle measures to prove that braces CG, BF, and AE are parallel.
Statement
Justification
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Your Turn
Use a different strategy to prove CG, BF and AE are parallel.
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Textbook page 78
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Key Ideas: • Angle Rules: When two parallel lines are cut by a transversal the following angle rules apply. • Alternate interior angles are congruent o ∠cae=∠aef, ∠dae=∠gea • Corresponding angles are congruent o ∠cab=∠gea, ∠bad=∠fea, ∠cae=∠geh, ∠dae=∠feh • Vertical angles are congruent o ∠cab=∠dae, ∠bad=∠cae, ∠gea=∠hef, ∠geh=∠fea • When a transversal intersects a pair of non‐parallel lines, the corresponding angles are not equal. • Remember that two angles that lie on a straight line are supplementary (add to 180o)
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Find the missing angles. Justify your answer.
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Find the missing angles. Justify your answer.
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Find the missing angles. Justify your answer.
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2.2AnglesFormedbyParallelLines.notebook
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Find the missing angles. Justify your answer.
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2.2AnglesFormedbyParallelLines.notebook
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Find the missing angles. Justify your answer.
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2.2AnglesFormedbyParallelLines.notebook
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Find the missing angles. Justify your answer.
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2.2AnglesFormedbyParallelLines.notebook
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Find the missing angles. Justify your answer.
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2.2AnglesFormedbyParallelLines.notebook
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Find the missing angles. Justify your answer.
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Find the measure of each angle in the diagram.
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Find the measure of each angle in the diagram.
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2.2AnglesFormedbyParallelLines.notebook
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Find the measure of each angle in the diagram.
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2.2AnglesFormedbyParallelLines.notebook
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Find the measure of each angle in the diagram.
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2.2AnglesFormedbyParallelLines.notebook
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Find the measure of each angle in the diagram.
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2.2AnglesFormedbyParallelLines.notebook
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Find the measure of each angle in the diagram.
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2.2AnglesFormedbyParallelLines.notebook
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Find the measure of each angle in the diagram.
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2.2AnglesFormedbyParallelLines.notebook
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Find the measure of each angle in the diagram.
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2.2AnglesFormedbyParallelLines.notebook
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Find the measure of each angle in the diagram.
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2.2 Assignment: Nelson Foundations of Mathematics 11, Sec 2.2, pg. 78‐82 Questions: 1 ‐ 5, 8, 10, 12, 15, 20
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