In the Geometry Textbook, Math 2 Warm Up In the Geometry Textbook, p. 177 #1-13 p. 178 #4, 6-9 Turn in after we check!!

Math 2 Warm Up

Math 2 Warm Up

Unit 2: Congruence & Similarity “Congruent Figures” Objective: To recognize congruent figures and their corresponding congruent parts. congruent figures: two or more figures (segments, angles, triangles, etc.) that have the “same shape” and the “same size”. symbol for congruent: ≅ congruent polygons: two polygons are congruent if all the pairs of corresponding sides and all the pairs of corresponding angles are congruent.

Examples of Congruent Polygons "Slide", "Flip", "Turn"...Translate, Reflection, Rotate If two figures are congruent, then one figure can be mapped onto the other one by a one or series of “rigid motions”!

How do I know if sides or angles are congruent? 1. If figures are drawn to scale, then measure the corresponding angles and measure the corresponding sides. 2. If figures are not drawn to scale, by special markings. Side Markings (“ticks”) Angle Markings (“hoops”)

“Reflexive Property of Congruence” Example: Given: ∆REM ≅ ∆FEM List the corresponding congruent parts. R E F M “Reflexive Property of Congruence” "If two figures share the same side or the same angle, then the shared sides or shared angles are congruent to each other."

Example: Given: ∆RTV ≅ ∆LTC List the corresponding congruent parts.

Example: Given: ∆RTV ≅ ∆CTL List the corresponding congruent parts.

m∠S = 36°, m∠Q = 27°, PR = 7 cm, and TS = 18 cm Find the m∠T. Example: Given: ΔPQR ≅ ΔSTU, m∠S = 36°, m∠Q = 27°, PR = 7 cm, and TS = 18 cm Find the m∠T. Find the m∠U. Find the SU. Find the QP.

Proving Triangles Congruent Example: Prove: ∆EAB ≅ ∆CDB .

Proving Triangles Congruent Example: Prove ∆PQR ≅ ∆PSR. “Third Angle Theorem” "If two angles of one triangle are congruent two angles of another triangle, then the third angles are congruent."

Proving Triangles Congruent “All right angles are congruent." Example: Prove ∆ABC ≅ ∆EBC . “Right Angle Theorem” “All right angles are congruent."

Given: AE ≅ DC, EB ≅ CB, B is the midpoint of AD, ∠E≅ ∠C Prove: ∆ABE ≅ ∆DBC

Example: Given: ∆ABC ≅ ∆QTJ List the corresponding congruent parts.

ΔABC ≅ ΔHFC F B A H C 1. List the corresponding congruent sides. 2. List the corresponding congruent angles.

End of Day 1 pp. 182 #3-12, 24-27, 31, 32, 38-41

Unit 2: Congruency & Similarity “Proving Triangles Congruent: Sides” Objective: To prove two triangles congruent by using the sides of the triangles.   NCTM Illuminations: Triangle Congruence

Side-Side-Side

Can you prove the two triangles congruent?

Side-Angle-Side

Can you prove the two triangles congruent?

Example 1 Given: M is the midpoint of 𝐗𝐘 , AX ≅ 𝐀𝐘 Prove: ∆AMX ≅ ∆AMY

Example 2 Given: RS bisects ∠GSH, SG ≅ 𝐒𝐇 Prove: ∆RSG ≅ ∆RSH R G H S

Example 3 Given: O is the midpoint of 𝐏𝐑 and 𝐄𝐖 Prove: ∆POW ≅ ∆ROE E P

Example 4 Given: 𝐑𝐒 ǁ 𝐓𝐊 , 𝐑𝐒 ≅ 𝐓𝐊 Prove: ∆RSK ≅ ∆TKS R K T S

Example 5* Given: 𝐀𝐁 is the perpendicular bisector of 𝐗𝐘 Prove: ∆AXB ≅ ∆AYB A X Y B

End of Day 2 pp 189 1-4, 12-17, 33, 41, 43

Unit 2: Congruency & Similarity “Proving Triangles Congruent: Angles” Objective: To prove two triangles congruent by using the angles of the triangles. NCTM Illuminations: Triangle Congruence

Angle-Side-Angle

Angle-Angle-Side

Can you prove the two triangles congruent? 1. Given: HI ≅ 𝐐𝐑 , ∠G ≅ ∠P, ∠H ≅ ∠Q 2. Given: ∠LPA ≅ ∠YAP, ∠LAP ≅ ∠YPA G H I L A Y P P R Q

Can you prove the two triangles congruent? 3. Given: ∠B ≅ ∠C, AX bisects ∠BAC 4. Given: TR ǁ AL A B C X Y R A T L

Example 5 Given: P is the midpoint of AB , ∠A ≅ ∠B Prove: ∆APX ≅ ∆BPY

Example 6 Given: PR bisects ∠SPQ, ∠S ≅ ∠Q Prove: ∆SRP ≅ ∆QRP R P Q S

Example 7 T Q R X M Given: XQ ǁ TR , X𝑹 bisects QT Prove: ∆XMQ ≅ ∆RMT

Can you prove the two triangles congruent? 3. Given: ∠B ≅ ∠C, AX bisects ∠BAC 4. Given: ∠YRT ≅ ∠YAL, ∠RTY ≅ ∠YLA A B C X Y R A T L

pp 197 #1-6, 10-15, 18, 29, 31, 32 End of Day 3

Unit 2: Congruency & Similarity “Proving Right Triangles Congruent” Objective: To prove two triangles congruent by using the sides of the triangles.   Hypotenuse Leg Leg

Hypotenuse-Leg Theorem 4-6 Hypotenuse – Leg (HL) If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent

Given: △PQR and △XYZ are right triangles, with right angles Q and Y respectively. Prove: △PQR ≅ △XYZ P X R Y Z

Given: CA ≅ ED AD is the perpendicular bisector of CE Prove: △CBA ≅ △EBD A C E D B

Given: WJ ≅ KZ ∠W and ∠K are right angles Prove: △JWZ ≅ △ZKJ Z W J K

Corresponding Parts of Congruent Triangles are Congruent “C.P.C.T.C.” We have used SSS, SAS, ASA, AAS, and HL to prove triangles are congruent. We also discussed the definition of congruent shapes (all corresponding parts of those shapes are also congruent). We will use the abbreviation CPCTC to say that Corresponding Parts of Congruent Triangles are Congruent. 1st Prove the triangles are congruent 2nd Use CPCTC for your reason the parts are congruent

D G F E Given: ∠EDG ≅ ∠EDF ∠DEG and ∠DEF are right angles Prove: EF ≅ EG D G F E

Given: 𝑆𝑃 ≅ 𝑃𝑂 ∠SPT ≅ ∠OPT Prove: ∠S ≅ ∠O O S T P

Given: 𝑃𝑅 ∥ 𝑀𝐺 and 𝑀𝑃 ∥ 𝐺𝑅 Prove: 𝑃𝑅 ≅ 𝑀𝐺 and 𝑀𝑃 ≅ 𝐺𝑅 R P M G

p 205 7-10, 14; p 221 16, 17 End of Day 4

Unit 2: Congruency & Similarity “Isosceles Triangles” Objective: To identify and apply properties of isosceles triangles. “legs” – are the two congruent sides. “base” – is the third non-congruent side. “vertex angle” – is the angle formed by the legs. “base angles” – are the angles formed using the base as a side.

Isosceles Triangle Theorem “If two sides of a triangle are congruent, then the angles opposite those sides are congruent.” If THEN

Example 1 Given: YX ≅ ZX and m∠YXZ = 30° Find the m∠ZYX and m∠YZX. Y Z

Example 2 Given: MO ≅ 𝐍𝐎 and m∠NMO = 50° Find the m∠MNO and m∠MON. M O

Converse Isosceles Triangle Theorem “If two angles of a triangle are congruent, then the sides opposite those angles are congruent.” If THEN

Example 3 Given: ∠R ≅ ∠T, RS = 5x – 8, ST = 2x + 7, RT = 4x + 2 Find x, RS, ST, and RT. S R T

Example 4 List any pair of segments that you would know are congruent.

Example 5 List any pair of segments that you would know are congruent.

Isosceles Bisector Theorem “The bisector of the vertex angle of an isosceles triangle is the perpendicular bisector of the base.” If THEN

Example 6 Z° 23° Find the value of x, y, and z.

Example 7 Find the value of x, y, and z. Z°

Equilateral Triangle Theorem “If all three sides of a triangle are congruent, then all three angles of the triangle are congruent.” If THEN

Converse Equilateral Triangle Theorem “If all three angles of a triangle are congruent, then all three sides of the triangle are congruent.” If THEN

Example 8 Find the value of w, x, y, and z.. X W y Z

Example 9 Find the measures of ... m∠BCA = ______ m∠BCD = ______ m∠DCE = ______ m∠BAG = ______ m∠DEF = ______ m∠GAH = ______

Example 10 Find the measures of ... m∠ABE = ______ m∠BAE = ______ m∠BDF = ______ m∠BDC = ______ m∠EFC = ______ m∠DFG = ______ m∠BEF = ______ m∠AEG = ______ m∠EBC = ______

p 213 #3-9, 21-26 End of day 5

Unit 2: Congruence & Similarity “Similarity in Right Triangles” Objective: To find and use the relationships in similar right triangles. geometric mean of two positive numbers a and b, is the positive number x such that 𝒂 𝒙 = 𝒙 𝒃 . Find the geometric mean of… 3 and 12 5 and 7

“Right Triangle Similarity” b c x y z Which segments are geometric means? What proportions can be set up?

Example 1 Solve for x, y, and z.

Example 2 Solve for x, y, and z.

Example 3 Solve for x, y, and z.

Example 4 Solve for x.

Example 5 Solve for x.

Side Splitter Theorem = if a line is parallel to a side of a triangle and intersect the other two sides, then this line divides those two sides proportionally. A AD AE DB EC = D E B C For proof: www.youtube.com/watch?v=6C2xHEGRTyl

Solve for x. 12 8 9 x

What is the length of side BC? X 5 B E X + 4 7 D C

Triangle Angle Bisector Theorem If a ray bisects an angle of a triangle, then it divides the opposite side into two segments that are proportional to the two other sides of the triangle. A BC BA CD DA = B C D For proof: https://www.youtube.com/watch?v=XLYUveKSCtY

A D C B 15 20 8 x Solve for x.

Solve for x. A D B C X-4 25 20

End of Day 6 pp. 442-443 #1-7 odd, 15-21, 34, 36

Unit 2: Congruence & Similarity “Similar Figures” Objective: To identify similar polygons, prove two triangles similar and use similar figures to find missing measurements. Two polygons are similar if, their corresponding angles are congruent and their corresponding sides are proportional (same ratio). The ratio of the lengths of corresponding sides is the similarity ratio.

Understanding Similarity Determine whether the triangles are similar. If they are, write a similarity statement and give the similarity ratio.

Practice!

Using Similarity

Example Given: LMNO ~QRST Find the value of x .

Angle-Angle Similarity

Side-Angle-Side Similarity If Then

Side-Side-Side Similarity If Then

Practice!

Example Use similar triangles to solve for x - if possible. 1. 2.

Example Use similar triangles to solve for x - if possible. 3. 4.

Example Use similar triangles to solve for x - if possible. 5.

Applying Similar Triangles In sunlight, a flagpole casts a 15 ft shadow. At the same time of day a 6 ft person casts a 4 ft shadow. Use similar triangles to find the height of the flag pole?

Applying Similar Triangles Brianna places a mirror 24 feet from the base of a tree. When she stands 3 feet from the mirror, she can see the top of the tree reflected in it. If her eyes are 5 feet above the ground, how tall is the tree?

Example Given: ∆ABC ~∆XYZ Complete each statement. a) b)

Example Given: ∆ABC ~∆XYZ Find the value of x .

End of Day 7 p 425 1-8, 13, 17; p 3-19 odds, 24-28