Unit 4 Congruent & Similar Triangles

Lesson 4.1 Day 1: Congruent Triangles

Lesson 4.1 Objectives Identify corresponding parts of congruent figures. Characterize congruent figures based on a congruence statement. Identify congruent triangles by using congruence theorems and postulates. (G2.3.1)

Congruent Triangles When two triangles are congruent, then –Corresponding angles are congruent. –Corresponding sides are congruent. Corresponding, remember, means that objects are in the same location. –So you must verify that when the triangles are drawn in the same way, what pieces match up?

Naming Congruent Parts Be sure to pay attention to the proper notation when naming parts. For instance: –  ABC   DEF By the way, this is called a congruence statement. »The order of the first triangle is usually done in alphabetical order. »The order of the second triangle must match up the corresponding angles. A B C D E F That way we know: and We also know: and

Example 4.1 In the figure above,  TJM   PHS. Complete the following statements. a)Segment JM  ___________ a)segment HS b)  P  ________ b)  T c)m  M  ________ c)48 o d)m  P = ________ d)73 o e)MT = ________ e)5 cm f)  HPS  __________ f)  JTM Yes, the order is important!

Postulate 19: Side-Side-Side Congruence Postulate If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent. –Abbreviated –SSS That means, if (S IDE ) and then

Postulate 20: Side-Angle-Side Congruence Postulate If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent. –Abbreviated –SAS That means, if (S IDE ) (A NGLE ) (S IDE ) and then

Example 4.2 Is there enough information given to prove the triangles are congruent? If so, state the postulate or theorem that would prove them so It is the same segment, just in two different triangles. But since it is the same segment, it has to be congruent. So that makes the third congruence we need to find the congruent triangles. When two lines intersect, they form vertical angles, and vertical angles are always congruent. So that makes the third congruence we need to find the congruent triangles. Yes, SSS No, SSA does not guarantee congruent  s. Yes, SAS No, SSA does not guarantee congruent  s.

Lesson 4.1a Homework Lesson 4.1: Day 1 – Congruent Triangles –p1-2 Due Tomorrow

Lesson 4.1 Day 2: More Congruent Triangles

Postulate 21: Angle-Side-Angle Congruence If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the two triangles are congruent. –Abbreviated –ASA That means, if (A NGLE ) (S IDE ) (A NGLE ) and then

Theorem 4.5: Angle-Angle-Side Congruence If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of the second triangle, then the two triangles are congruent. –Abbreviated –AAS That means, if (A NGLE ) (S IDE ) and then

Example 4.3 Is there enough information given to prove the triangles are congruent? If so, state the postulate or theorem that would prove them so It is the same segment, just in two different triangles. But since it is the same segment, it has to be congruent. So that makes the third congruence we need to find the congruent triangles. When two lines intersect, they form vertical angles, and vertical angles are always congruent. So that makes the third congruence we need to find the congruent triangles. Yes, ASA Yes, AAS No, there needs to be at least one pair of congruent sides.

Theorem 4.8: Hypotenuse-Leg Congruence Theorem If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of a second right triangle, then the two triangles are congruent. –Abbreviate using HL –It still means, if (H YPOTENUSE ) (L EG ) (R IGHT A NGLE ) with then

Example 4.4 Determine if enough information is given to conclude the triangles are congruent using HL Congruence? It is the same segment, just in two different triangles. But since it is the same segment, it has to be congruent. So that makes the third congruence we need to find the congruent triangles. Yes No, because HL Congruence only works in right triangles.

Which One Do I Use?…How Can I Tell? If there are more sides marked than angles, then use one of the following: –SSS If ALL three sides are marked as congruent pairs. –SAS If ONLY two sides are marked with the angle IN BETWEEN. –HL Only works in RIGHT TRIANGLES –Notice the angle is NOT IN BETWEEN the two sides. »It looks like SSA. If there are more angles marked than sides, then use one of the following: –ASA If ANY two angles are marked with the side IN BETWEEN. –AAS If ONLY two angles are marked along with a side that is NOT IN BETWEEN. –Notice, the side must come NEXT to the same angle that is marked the same way in both triangles. The best way to determine which congruence postulate/theorem is to identify the number of each part that is marked.

Lesson 4.1b Homework Lesson 4.1: Day 2 – More Congruent Triangles –p3-4 Due Tomorrow

Lesson 4.2 Proving Triangles are Congruent

Lesson 4.2 Objectives Create a proof for congruent triangles. (G2.3.1) Identify corresponding parts of congruent triangles. (G2.3.2)

Remembering Proofs Do you remember how to write a two-column proof? What is the first step? Rewrite the problem because… That was what was GIVEN to you. What should you write in the left-hand column? –The STATEMENTS you are making as you try to solve the problem. What must you show in the right-hand column? –The REASONS for why you made that statement.

The Understood Properties of Congruence PropertyReflexiveSymmetricTransitive Definition Why use this property? This property is used to show that a segment is shared between two figures. This property allows us to change sides of an equal or congruent sign without losing the truth of the statement. This property allows us to conclude that when multiple segments are congruent to the same segment, then they are congruent to each other. How it’s identified in a picture Usually in the proof itself, not in the picture. Other places it’s used IF AND THEN

Example 4.5

Example 4.6 Any time you are GIVEN a vocabulary word… You should DEFINE that term in your proof by… SHOWING what the term does to your picture.

Proof Building Tips Here are a few helpful hints to building a proof: 1.If there is a mark already on the picture, then there should be a step in the proof to explain the mark. 1.Those marks should be from the GIVEN, but if not…the reason in the proof would still be GIVEN. 2.Any time you add a mark to the picture, you need a step for that it in your proof. 2.This would be a good time for the REFLEXIVE PROPERTY. –Or things like »Vertical Angles »Midpoint »Parallel Line Theorems »Transitive Property (MAYBE) 3.Always be sure the last STATEMENT in the proof is an exact match to what you are trying to PROVE in the problem.

Example 4.7 Once you have 3 congruencies, you should have enough to prove the triangles are congruent.

Lesson 4.2 Homework Lesson 4.2 – Proving Congruent Triangles –p5-6 Due Tomorrow

Lesson 4.3 Similar Triangles

Lesson 4.3 Objectives Show triangles are similar using the correct postulate/theorem. (G2.3.3) Solve similar triangles. (G2.3.4) Utilize the scale factor and proportions to solve similar triangles. (G2.3.5)

Ratio If a and b are two quantities measured in the same units, then the ratio of a to b is a / b. –It can also be written as a:b. A ratio is a fraction, so the denominator cannot be zero. Ratios should always be written in simplified form. – 5 / 10  1 / 2

Similarity of Polygons Two polygons are similar when the following two conditions exist – Corresponding angles are congruent. – Corresponding sides are proportional. Means that all corresponding sides fit the same ratio. Basically we are saying we have two polygons that are the same shape but different size. We use similarity statements to name similar polygons. –GHIJ ~ KLMN You must match the order of the second polygon with that of the first to show corresponding angles and sides!

Scale Factor Remember, polygons will be similar when their corresponding sides all fit the same ratio. That common ratio is called a scale factor. –We use the variable k to represent the scale factor. So k = 3 / 2 or k = 2 / 3, depending on which way you look at it. »Remember, it’s a ratio!

Postulate 25: Angle-Angle Similarity Postulate If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar.

Theorem 8.2: Side-Side-Side Similarity If all the corresponding sides of two triangles are proportional, then the triangles are similar. –Your job is to verify that all corresponding sides fit the same exact ratio! Shortest sides2 nd shortest sidesLongest sides

Theorem 8.3: Side-Angle-Side Similarity If an angle of one triangle is congruent to an angle of a second triangle and the lengths of the corresponding sides including these angles are proportional, then the triangles are similar. –Your task is to verify that two sides fit the same exact ratio and the angles between those two sides are congruent! SIDE ANGLE

Example Are the triangles similar? 2.If so, list the congruent angles. 3.Also, what is the scale factor? YES, by SSS

Example 4.9 State if the triangles are similar. If so, write a similarity statement AND the postulate or theorem that makes them similar VA Yes, by SAS Similarity. VA Yes, by AA. Yes, by SSS Similarity.

Example 4.10 State if the triangles are similar. If so, write a similarity statement AND the postulate or theorem that makes them similar C A E A D B Reflexive Property Corresponding Angles Postulate Yes, by AA Similarity Not similar Scale factors must be equal!

Lesson 4.3 Homework Lesson 4.3 – Similar Triangles –p7-8 Due Tomorrow

Lesson 5.4 Using Similar Triangles

Lesson 4.4 Objectives Identify corresponding parts of congruent figures. Solve similar triangles. (G2.3.4) Utilize the scale factor and proportions to solve similar triangles. (G2.3.5)

Proportion An equation that has two ratios equal to each other is called a proportion. –A proportion can be broken down into two parts: Extremes –Identifies the partnership of the numerator of the first ratio and the denominator of the second ratio. Means –Identifies the partnership of the denominator of the first ratio and numerator of the second ratio. b ca d =

Solving Proportions To solve a proportion, you must use the cross product property. –More commonly referred to as cross-multiplying. So multiply the extremes together and set them equal to the means. b ca d = MULTIPLY ad = bc

Example 4.11 Solve the following proportions using the Cross Product Property

Solving Similar Triangles You must use a proportion to solve similar triangles. To set-up the proportion it is best to match corresponding sides in each ratio. –BE CAREFUL TO SET UP EACH RATIO THE SAME WAY So make the top of each ratio represent the small triangle (for instance) and the bottom of each ratio represent the larger triangle.

Example 4.12 Find the scale factor. Largest sides must work together. Or partner the smallest sides together IN THE SAME ORDER. Either way, it should result in the same ratio for similar triangles.

Example 4.13 Find KJ. Notice, each ratio was made with corresponding parts in the SAME ORDER!

Example 4.14 Solve for x using the given scale factor. The other ratio comes from using the scale factor.

Example 4.15 Find the height, h, of the flagpole. Notice, h is a part of the large triangle. So be careful when selecting the length of another side when building your proportion.

Lesson 4.4 Homework Lesson 4.4 – Using Similar Triangles –p9-10 Due Tomorrow