DAY 1 Parallel Lines Transversals Parallel Lines Never
- Slides: 54
DAY 1 – Parallel Lines & Transversals
Parallel Lines • Never intersect • Have a constant distance between them • Have the same slope
Transversal • A transversal cuts two or more parallel lines at two distinct points. • Allows for special pairs of angles to form. • An angle breaks up a plane into three regions.
Regions created by an angle • EXTERIOR of the angle • INTERIOR of the angle • ON the angle
Alternate Exterior • Alternate: Opposite • Exterior: Outside • Theorem: Pairs of alternate exterior angles are congruent. • Angles 1 & 8
Alternate Interior • Alternate: Opposite • Interior: Inside • Theorem: Pairs of alternate interior angles are congruent. • Angles 4 & 5
Consecutive Exterior • Consecutive: Next to; One after the other • Exterior: Outside • Theorem: Pairs of consecutive exterior angles are supplementary. • Angles 1 & 7
Consecutive Interior • Consecutive: Next to; One after the other • Interior: Inside • Theorem: Pairs of consecutive exterior angles are supplementary. • Angles 3 & 5
Corresponding • Corresponding: Equivalent; the same • Theorem: Pairs of corresponding angles are congruent. • Angles 3 & 7
Angle 6 – Vertical Angle 3 – Corresponding Angle 8 – Alternate Exterior Angle CHECK FOR UNDERSTANDING
WHITE BOARD WORK • Solve for X. • Be ready to explain your work. • Have reason for your answer
WHITE BOARD WORK • Find the measurement of angle ABD. • Does your answer make sense?
Solve for x in more than one way. CHECK FOR UNDERSTANDING
Work. Session
DAY 2 – Congruent Triangles
Congruence in Triangles • Two triangles are congruent if and only if their corresponding angles and sides are congruent. • Corresponding parts: matching parts/sides/angles • Corresponding Parts of Congruent Triangles are Congruent (CPCTC) • Later we will use CPCTC as part of our proofs.
Included Parts • Included Side o The side between two angles o Segment IG • Included Angle o The angle between two sides o Angle A
Extra Markings • Share a side o Reflexive Property o Segment MP • Vertical Angles o Vertical Angles are Congruent o Angles NSP & UST
Side-Side • Three sides of 1 triangle are congruent to three sides of another triangle, the triangles are congruent. • White. Boards: Congruent Sides
Side-Angle-Side • If two sides and the included angle of one triangle are congruent to two sides and the included angle of another, the triangles are congruent.
Angle-Side-Angle • If two angles and the included side are congruent to two angles and the included side of another triangle, then the two triangles are congruent.
Angle-Side • If two angles and a nonincluded side of one triangle are congruent to the corresponding two angles and nonincluded side, then the two triangles are congruent.
Hypotenuse-Leg • If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent.
CHECK FOR UNDERSTANDING • Mark the appropriate sides and angles to make each congruence statement true by the stated congruence theorem.
DAY 3 – Similarity and Proportions
Magic Mirror Legend has it that if you stare into a person’s eyes in a special way, you can hypnotize them into squawking like a chicken. 1. Your victim has to stand exactly 200 cm away from a mirror and stare into it. 2. The only tricky part is that you need to figure out where you have to stand so that when you stare into the mirror, you are also staring into your victim’s eyes. 3. If your calculations are correct and you stand at the correct distance, you will be able to stare directly into the victim’s eyes.
THINK: Take 3 minutes to think about why this worked? PAIR: Take 3 minutes to pair with your partner. Compare your thoughts and revise if necessary. SHARE: Be prepared to share you & your partners thoughts with the class. Magic Mirrors: Why were we able to see in each others’ eyes no matter what the heights were?
RATIOS & PROPORTION S Comparison of two quantities. • Proportion • An equation stating two ratios are equal The Cross-Product • •
SIMILARITY • Figures that are similar have the same shape, but not necessarily the same size. • In order to prove figures are similar, their corresponding angles must be congruent and their corresponding sides must be proportional. • If two figures are congruent, they are always similar. • If two figures are similar, they are not congruent.
Side-Side Similarity Statement
Side-Angle-Side Similarity Statement
• If two angles of one triangle are congruent to two angles of another triangle, the triangles are similar. Angle-Angle Similarity Statement • Explanation: The sides of the triangle can extend, making one triangle larger than the other (Proportional).
CHECK FOR UNDERSTANDING
Review for Quiz 2. 1 -2. 2 Parallel Lines & Transversal Congruent Triangles Corresponding Parts
• Are these triangles congruent? • If so, how? • Which two ways can you NOT prove triangles to be congruent? Congruent Triangles
Side-Angle-Side
Side-Angle-Side
Find the measures of the angles. 40 140 140 40
DAY 4 – DILATIONS & SCALE FACTORS
DILATION • A proportional enlargement or reduction of a figure. • Center of Dilation: the point a figure if enlarged/reduced through • Scale Factor: the size of the enlargement/reduction. • Enlargement – greater than 1 • Reduction – less than 1 • A figure and its dilated image are always similar.
Congruency Similarity Congruency Check for understanding
I-Do Used a scale factor of 2. Complete the table below and graph both the original (pre-image) and new (image) rectangle. How did the following change? Pre-Image A (-2, -1) 2(-2, -1) B (2, -1) 2(2, -1) C (-2, 1) 2(-2, 1) D (2, 1) 2(2, 1) A. Angle Measures: THEY DON’T CHANGE B. Length of Sides: THEY ARE EACH LARGER BY 2 UNITS Process Image A’ ( , -4 -2 ) B’ ( , 4 -2 ) C’ ( , -4 2 ) D’ ( , 4 2 )
You-Do Used a scale factor of 1/2. Complete the table below and graph both the original (pre-image) and new (image) rectangle. How did the following change? Pre-Image A (-4, 4) B (-4, -4) A. Angle Measures: THEY DON’T CHANGE B. Length of Sides: THEY ARE EACH SHORTER BY 2 UNITS Process . 5(-4, 4). 5(-4, -4) C (4, 4) . 5(4, 4) D (4, -4) . 5(4, -4) Image A’ ( , -2 2 ) B’ ( , -2 ) 2 C’ ( , ) 2 2 D’ ( , ) 2 -2
Finding Scale Factors •
Example 3: •
Example 4: •
Center of Dilation • A fixed point in the plane about which all points in a figure are enlarged/reduced. • How to find: 1. Connect each corresponding vertex from the pre-image to the image. 2. The lines all meet at the center of dilation.
Finding Scale Factor • How to find the scale factor: 1. Find the center of dilation. 2. Compare the lengths/distances between the corresponding points to the center of dilation from new to old figure.
Performing Dilations How to perform dilations: 1. Find location from center of dilation. (R, L, U, D) 2. Multiply by the scale factor. 3. Find new locations. Move from center of dilation. Example 6 A: dilation by c = ¼, center (0, 0) Pre-Image Process Image
Performing Dilations Example 6 B: dilation by c = 2, center (0, 0) Pre-Image Process Image
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