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Secondary 2 Chapter 3/4

Secondary IIUnit 3 – Properties of Triangles

Unit 4 – Similarity through Transformations 2014/2015

Date Section Assignment

Concept

A: 9/23B: 9/24 3.1/3.2 - Worksheet 3.1 & 3.2

Triangle Sum, Exterior Angle, and Exterior Angle Inequality TheoremsThe Triangle Inequality Theorem

A: 9/25B: 9/26 3.3/3.4 - Worksheet 3.3 & 3.4

Properties of a 45-45-90 TriangleProperties of a 30-60-90 Triangle

A: 9/29B: 9/30 4.1/4.2 - Worksheet 4.1 & 4.2

Dilating Triangles to Create Similar TrianglesSimilar Triangle Theorems

A: 10/1B: 10/2 4.3/4.4 - Worksheet 4.3 & 4.4

Theorems About ProportionalityMore Similar Triangles

A: 10/3B: 10/6 4.5/4.6 - Worksheet 4.5 & 4.6

Proving the Pythagorean Theorem and the Converse P. TheoremApplication of Similar Triangles

A: 10/7B: 10/8 Review Worksheet

A: 10/9B: 10/10 Chapter 3/4 TEST

Late and absent work will be due on the day of the review (absences must be excused). The review assignment must be turned in on test day. All required work must be complete to get the curve on the test.

Remember, you are still required to take the test on the scheduled day even if you miss the review, so come prepared. If you are absent on test day, you will be required to take the test in class the day you return. You will not receive the curve on the test if you are absent on test day unless you take the test prior to your absence.

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Chapter 3/4: Probability and Counting

3.1/3.2 – Triangle Sum, Exterior Angle, Exterior Angle Inequality, and Triangle inequality Theorems

The Triangle Sum Theorem states: The sum of the measures of the interior angles of a triangle is 180°.

Example 1: Draw an acute scalene triangle. Use a protractor to measure each interior angle and label the angles measures.

1. Measure the length of each side of the triangle. Label the sides in your diagram.

2. Which interior angle is opposite the longest side of the triangle?

3. Which interior angle lies opposite the shortest side of the triangle?

Example 2: List the sides from shortest to longest in each diagram.

a. b. c.

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Example 3: Use the diagram shown to answer the following questions.

1. Name the interior angles of the triangle. 2. Name the exterior angles of the triangle.

3. What does m∠1+m∠2+m∠3 equal? Explain.

4. What does m∠3+m∠ 4 equal? Explain.

5. Why does m∠1+m∠2=m∠4?

The remote interior angles of a triangle are the two angles that are non-adjacent to the specified exterior angle.

The Exterior Angle Theorem states: The measure of the exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles of a triangle.

Example 4: Solve for x in each diagram.

a. b.

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c. d.

The Exterior Angle Inequality Theorem states: The measure of the exterior angle of a triangle is greater than the measure of either of the remote interior angles of the triangle.

Example 4: In groups, roll a dice 3 times to get values for three potential sides of a triangle. Write each of the three rolls on the chart below. Once you have written them down, using a ruler and the space below, determine whether the three sides form a triangle.

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Roll 1 Roll 2 Roll 3 Forms a triangle?


1. Compare the lengths that formed a triangle with the ones that did not. What you do notice?

2. Under what conditions were you able to form a triangle?

3. Under what conditions were you unable to form a triangle?

Example 5: Determine if it is possible to form a triangle using segments with the following measurements. Explain your reasoning.

a. 2 cm, 5.1 cm, 2.4 cm b. 9.2 cm, 7 cm, 1.9 cm

The Triangle Inequality Theorem states: The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

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Additional Notes

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3.3/3.4 – Properties of 45-45-90 and 30-60-90 Triangles

Rationalizing the Denominator

Square roots are NOT in simplest form when there is a square root in the denominator. Rationalizing the denominator is the process used to get rid of the square root in the denominator. To do this, simplify the square root, and then multiply the numerator and denominator by the simplified square root.

Example 1: Simplify the following radicals.

a. b. √ 13

c.

2√15√3 d.

32√5

You will frequently need to determine the value of trigonometric ratios for - , and angles. In

Trigonometry we study the special triangle and the special triangle.

Simplest Side Ratios for Special Right Triangles

30° 45°

√3 2 1 √2

60° 45° 1 1

Example 1: Find the missing sides using the Special Right Triangles.

a. b.

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c. d. e.

1. Explain how you calculate the length of the hypotenuse given a leg.

2. Explain how you calculate the length of the side given the hypotenuse.

Example 2: Find the missing sides using the Special Right Triangles.

a. b. c.

d. e. f.

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Explain how you calculate the following on a triangle:

1. The length of the hypotenuse given the length of the shorter leg.

2. The length of the hypotenuse given the length of the longer leg.

3. The length of the shorter leg given the length of the longer leg.

4. The length of the shorter leg given the length of the hypotenuse.

5. The length of the longer leg given the length of the shorter leg.

6. The length of the longer leg given the length of the hypotenuse.

Example 3: Find the missing side using the Special Right Triangles.

a. b.

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Additional Notes

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4.1/4.2 – Dilating Triangles to Create Similar Triangles, and Similar Triangle Theorems

Example 1: Redraw the given figure such that it is twice its size.

1. When you redrew the figure, did the shape of the figure change?

2. When you redrew the figure, did the size of the figure change?

3. How did you get the measurements for the redrawn figure?

4. What is the ratio of the short side of the smaller figure to the short side of the larger figure?

5. What is the ratio of the long side of the smaller figure to the long side of the larger figure?

6. What do you notice about the ratios?

7. What do you notice about the corresponding angles in each of the figures?

8. What is the relationship between the image and pre-image?

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Because the second figure was twice as big as the given figure, it is considered to have a dilation factor of 2.

Similar Triangles are triangles that have all pairs of corresponding angles congruent and all corresponding sides proportional. Similar triangles have the same shape but not always the same size.

Example 2: Triangle J ' K ' L' is a dilation of ∆ JKL. The center of the dilation is the origin.

1. List the coordinates of the vertices of ∆ JKL and ∆ J ' K ' L' . How do the coordinates of the image compare to the coordinates of the pre-image?

2. What is the scale factor of the dilation? Explain.

3. How do you think you can use the scale factor to determine the coordinates of the vertices of an image?

4. Use coordinate notation to describe the dilation of point (x , y ) when the center of the dilation is at the origin using a scale factor of k.

Example 3: Triangle HRY Triangle JPT . Draw a diagram that illustrates this similarity statement and list all of the pairs of congruent angles and all of the proportional sides.

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Example 4: What conditions are necessary to show triangle GHK is similar to triangle MHS?

The Angle-Angle Similarity Theorem states: If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

If m∠ A=m∠D and m∠C=m∠F, then ∆ ABC ∆≝.

Example 5: The triangles shown are isosceles triangles. Do you have enough information to show that the triangles are similar? Explain your reasoning.

Example 6: The triangles shown are isosceles triangles. Do you have enough information to show that the triangles are similar? Explain your reasoning.

The Side-Side-Side Similarity Theorem state: If all three corresponding sides of two triangles are proportional, then the triangles are similar.

If ABDE

= BCEF

= ACDF

, then∆ ABC ∆≝.

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Example 6: Determine whether ∆UVW is similar to ∆ XYZ. If so, use symbols to write a similarity statement.

An included angle is formed by two consecutive sides of the figure.

An included side is a line segment between two consecutive angles.

The Side-Angle-Side Similarity Theorem states: If two of the corresponding sides of two triangles are proportional and the included angles are congruent, then the triangles are similar.

If ABDE

= ACDF and ∠ A≅∠D, then ∆ ABC ∆≝.

Example 6: Determine whether the pair of triangles are similar. Explain your reasoning.

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Additional Notes

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4.3/4.4 – Theorems about Proportionality and More Similar Triangles

The Angle Bisector/Proportional Side Theorem states: A bisector of an angle in a triangle divides the opposite side into two segments whose lengths are in the same ratio as the lengths of the sides adjacent to the angle.

If AD bisects ∠BAC, then ABAC

=BDCD

Example 1: On the map shown, North Craig Street bisects the angle formed between Bellefield Avenue and Ellsworth Avenue.

The distance from the ATM to the Coffee Shop is 300 feet. The distance from the Coffee Shop to the Library is 500 feet. The distance from your apartment to the Library is 1200 feet.

1. Determine the distance from you apartment to the ATM.

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Example 2: CD bisects ∠C. Solve for DB.

Example 3: CD bisects ∠C. Solve for AC.

Example 4: AD bisects ∠ A. AC+AB=36. Solve for AC and AB.

Example 5: BD bisects ∠B. Solve for AC.

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The Triangle Proportionality Theorem states: If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally.

If BC is parallel to DE, then BDDA

=CEEA .

Example 6: Use the Triangle Proportionality Theorem to determine the missing value.

The Converse of the Triangle Proportionality Theorem states: If a line divides the two sides of a triangle proportionally, then it is parallel to the third side.

If BDDA

=CEEA , then BC is parallel to DE.

The Proportional Segments Theorem states: If three parallel lines intersect two transversals, then they divide the transversals proportionally.

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If L1∥L2 ∥L3, then ABBC

=DEEF

.

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Example 6: Use the Proportional Segments Theorem to determine the missing value.

The Triangle Midsegment Theorem states: The midsegment of a triangle is parallel to the third side of the triangle and is half the measure of the third side of the triangle.

If JG is the midsegment of the triangle, then

JG∥DS , and JG=12DS .

Example 7: Ms. Zoid asked her students to determine whether RD is a midsegment of ∆TUY , given TY=14 cm andRD=7cm. Carson told Alicia that using the Midsegment Theorem, he could conclude that RD is a midsegment. Is Carson correct? Explain.

Example 8: Ms. Zoid drew a second diagram on the board and asked her students to determine if RD is a midsegment of ∆TUY , given RD∥TY . Alicia told Carson that using the Midsegment Theorem, she could conclude that RD is a midsegment. Is Alicia correct? Explain.

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The altitude of a triangle is the line segment drawn to a vertex of a triangle perpendicular to the line containing the opposite side.

CD is the altitude of AB.

The Right Triangle Altitude Similarity Theorem states: If an altitude is drawn to the hypotenuse of a right triangle, then the two triangle formed are similar to the original triangle and to each other.

If CD is the altitude of AB, then ∆ ABC ∆ ACD ∆CBD.

The geometric mean of two positive numbers a and b is the positive number x such that ax= xb

The Right Triangle Altitude/Hypotenuse Theorem states: The measure of the altitude drawn from the vertex of the right angle of a right triangle to its hypotenuse is the geometric mean between the measures of the two segments of the hypotenuse.

If CD is the altitude of AB, then ADCD

=CDBD .

The Right Triangle Altitude/Leg Theorem states: If the altitude is drawn to the hypotenuse of a right triangle, each leg of the right triangle is the geometric mean of the hypotenuse and the segment of the hypotenuse adjacent to the leg.

If CD is the altitude of AB, then ACCD

=CDCB .

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Example 9: Solve for x in each triangle below.

a. b.

c. d.

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Additional Notes

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4.5/4.6 – Proving the Pythagorean Theorem and its Converse, Application of Similar Triangles

Example 1: Use the Right Triangle/Altitude Similarity theorem to prove the Pythagorean Theorem.

Given: Triangle ABC with right angle C

Prove: AC2+CB2=AB2

1. Construct altitude CD to hypotenuse CD.

2. Applying the Right Triangle/Altitude Similarity Theorem, what can you conclude?

3. Write a proportional statement describing the relationship between the longest leg and hypotenuse of triangle ABC and triangle CBD.

4. Rewrite the proportional statement you wrote in Question 3 as a product.

5. Write a proportional statement describing the relationship between the shortest leg and hypotenuse of triangle ABC and triangle ACD.

6. Rewrite the proportional statement you wrote in Question 5 as a product.

7. Add the statement in Question 4 to the statement in Question 6.

8. Factor the statement in Question 7.

9. What is equivalent to DB+AD?

10. Substitute the answer to Question 9 into the answer in Question 8 to prove the Pythagorean Theorem.

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Example 2: You go to the park and gather enough information to calculate the height of one of the trees. The figure shows your measurements. Calculate the height of the tree.

Example 3: Stacey lines herself up with the tree’s shadow so that the tip of her shadow and the tip of the tree’s shadow meets. A diagram is shown below. Calculate the height of the tree.

Example 4: You stand on one side of the creek and your friend stands directly across the creek from you on the other side as shown in the figure. Your friend is standing 5 feet from the creek and you are standing 5 feet from the creek. You and your friend walk away from each other in opposite parallel directions. Your friend walks 50 feet and you walk 12 feet.

1. Label and angle relationships that you know on the diagram. Explain how you know these angle measures.

2. How do you know that they triangle formed by the lines are similar?

3. Calculate the distance from your friend’s starting point to your side of the creek. Round your answer to the nearest tenth, if necessary.

4. What is the width of the creek?

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Additional Notes

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