Chapter 4 – Congruent Triangles
Section 1 – Classifying Triangles
- I can identify and classify triangles by angle measures and side measures.
Naming Triangles by Angles:
- Acute Triangle – a triangle with 3 acute angles inside it.
- Equiangular Triangle - a triangle with 3 congruent acute angles
- Obtuse Triangle – a triangle with 1 obtuse angle (the other 2 angles will be acute)
- Right Triangle - a triangle with 1 right angle (the other 2 angles will be acute)
1A. Classify the triangles as acute, equiangular, obtuse or right.
a. b.
2A. Classify XYZ as acute, equiangular, obtuse, or right. Explain your reasoning.
2B. Classify PQS as acute, equiangular, obtuse or right.
3. The triangular truss below is modeled for steel construction. Classify JMN, JKO, and OLN as acute, equiangular, obtuse, or right.
Naming Triangles by Sides:
- Equilateral Triangle – a triangle with 3 congruent sides.
- Isosceles Triangle – a triangle with at least 2 congruent sides (it can have 3 congruent sides, but MUST have at least 2). 2 sides will be congruent or equal and the side that is not equal is called the base.
- Scalene Triangle - a triangle with NO congruent sides.
4. If point M is the midpoint of JL, classify each triangle as equilateral, isosceles, or scalene. Explain your reasoning.
(a) JMK(b) KML(c) JKL
5. If point Y is the midpoint of VX, and WY = 3.0 units, classify VWY as equilateral, isosceles, or scalene. Explain your reasoning.
6. Find the measures of the sides of isosceles triangle KLM with base KL.
7. Find x and the measures of each side of equilateral triangle ABC if AB = 6x – 8, BC=7 + x, and AC = 13 – x.
***Do example 44 or 46 in class!!!!!
Homework – Page 239 – 241 (15 – 35 ODD, 36, 37, 40, 41, 42, 43, 45, 49, 51, 56, 60)
Section 4.2 – Angles of Triangles
- I can apply the Triangle-Sum Theorem and the Exterior Angle Theorem.
- I can prove the sum of the measures of the interior angles of a triangle is equal to 180
- Triangle Angle-Sum Theorem – The sum of the measures of the angles of a triangle is 180.
(ex: A + B + C = 180)
***GO over proofs in chapter!!
- The diagram shows the path of the softball in a drill developed by four players. Find the measure of each numbered angle.
1 (A and B): Find the measures of each numbered angle:
A.
B.
- Exterior Angle Theorem – The measure of a triangle is equal to the sum of the measures of the two remote interior angles. (ex: 1 + 3 = 4)
- Exterior Angle – an angle formed by one side of the triangle and the extension of an adjacent side (ex:4)
- Remote Interior Angles - 2 angles inside the triangle that are not adjacent to the exterior angle (ex:1 and 3)
2. Find the value of x and the measure of FLW in the fenced flower garden shown.
- Find the measure of each numbered angle.
****DO example problems out of the book (Guided practice on page 248)
Homework – Page 248 – 250 (16, 13 – 31 ODD, 32, 36, 38, 42, 47, 49)
Section 4.3 – Congruent Triangles
- I can explain and prove that in a pair of congruent triangles, corresponding sides are congruent and corresponding angles are congruent.
- Definition of Congruent Polygons CPCTC - (Stands for: Corresponding Parts of Congruent Triangles Corollary) 2 polygons are congruent if and only if their corresponding parts are congruent. These corresponding parts include corresponding angles and corresponding sides. ORDER MATTERS
- Show that the polygons are congruent by identifying all of the congruent corresponding parts. Then write a congruence statement.
1 (A and B): Same directions as above:
A. B.
2(a). In the diagram, ITP NGO. Find the values of x and y.
2(b). In the diagram, RSV TVS. Find the values of x and y.
- Third Angles Theorem – If 2 angles of 1 triangle are congruent to 2 angles of a second triangle, then the third angles of the triangles are congruent.
(ex: If C K and B J and then A L)
3. (a) A drawing of a tower’s roof is composed (b) In the diagram, if WNX = WRX,
of congruent triangles all converging at a point WX bisects NXR, mWNX = 88, and
at the top. If J K and m J = 72, find m JIH. mNXW = 49, find mNWR.
4.
Homework – Page 257 – 260 (10, 12, 13 – 16, 18 – 20, 23, 24, 28 – 30, 37 ALL)
Section 4.4: Proving Triangles Congruent – SSS and SAS
Section 4.5 – Proving Triangle Congruent – ASA and AAS
- I can use the definition of congruence, based on rigid motion, to explain the triangle congruence criteria (ASA, SAS, SSS, AAS, HL)
- I can use congruence criteria to solve problems about triangles and prove relationships in geometric figures.
- Side-Side-Side (SSS) Congruence – If 3 sides of one triangle are congruent to 3 sides of a second triangle, then the triangles are congruent.
(ex: If side AB DE, side BC EF, and AC DF, then ABC DEF)
- Side-Angle-Side (SAS) Congruence – If 2 sides and the included angle of 1 triangle are congruent to 2 sides and the included angle of a second triangle, then the triangles are congruent.
(ex: If side AB DE, B E, and side BC EF, then ABC DEF)
- Angle-Side-Angle (ASA) Congruence – If 2 angles and the included side of 1 triangle are congruent to 2 angles and the included side of another triangle, then the triangles are congruent.
(ex: If A D, side AB DE, and BD, then ABC DEF)
- Angle-Angle-Side (AAS) Congruence – If 2 angles and the nonincluded side of one triangle are congruent to the corresponding 2 angles and nonincludedside of a second triangle, then the 2 triangles are congruent.
(ex: If A D, B E, and side BC EF, then ABC DEF)
1.Use the following information to solve:
(a)Graph both triangles on the same coordinate plane.
(b)Use your graph to make a conjecture as to whether the triangles are congruent.
(c)Write a logical argument using coordinate geometry to support the conjecture you made in part b.
Triangle ABC with A(1,1), B(0,3) and C(2,5). Triangle EFG with E(1, -1), F(2, -5) and G(4, -4).
2. Triangle DVW with D(-5, -1), V(-1, -2), and W(-7, -4). Triangle LPM with L(1, -5), P(2, -1) and M(4, -7)
3 – 7 (a-e): Determine which postulate can be used to prove that the triangles are congruent. If not possible to prove congruence, write not possible.
3. 4.
5.6.
8. BCD WXY. Solve for x.
9. MHJ PQJ. Solve for y.
Homework – Page 267 - 269 (9, 11, 16 – 19, 27. 28, 31) ALL
Page 277 – 278 (14, 15) and Page 282 (7-9)ALL
Section 4.4 and 4.5 Proofs
1 – 8: Write a two-column proof for each of the following:
1. Given:
Prove:
2. Given: and G is the midpoint of both and
Prove:
3. Given: and
Prove:
4. Given: and
Prove:
5. Given: L is the midpoint of and
Prove:
6. Given: and
Prove:
7. Given: ,:, and
Prove:
8. Given: and
Prove:
Homework – Write ALL as two-column proofs Page 267 - 269 (5, 6, 12, 13)
Page 277 – 278 (6, 7, 9, 10, 17) and Page 282 (7-9)ALL
Section 4.6 – Isosceles and Equilateral Triangles
- I can apply the properties of isosceles and equilateral triangles to solve triangle problems.
- I can prove the base angles of isosceles triangles are congruent.
- Isosceles Triangle – Triangle with 2 congruent sides (the congruent sides are called the “legs”) and the included angle of the legs is called the vertex. The other side is called the base and the 2 angles across from the congruent sides are called the base angles.
- Refer to the following figure to answer questions:
(a)Name 2 unmarked congruent angles(b) Name 2 unmarked congruent sides
- Find the measure of each:
(a)mR(b) PR
- Find the value of each variable:
- Find the value of each variable:
Homework – Page 287 – 290 (9, 11, 13, 15– 22, 29 – 32, 38, 39, 48)
4.8– Triangles and Coordinate Proof
- I can write coordinate proofs by using coordinate geometry.
- I can represent the vertices of a figure in the coordinate plane using variables.
Steps for Placing Triangles on Coordinate Plane
- Use the origin as a vertex or center of the triangle.
- Place at least one side of a triangle on an axis.
- Keep the triangle within the first quadrant if possible.
- Use coordinates that make transformations as simple as possible.
- Position and label a right triangle XYZ with leg XZ d units long on the coordinate plane.
- Name the missing coordinates of isosceles right triangle ORS.
A.B.
B.
Homework – Page 305 (7 – 18) ALL