This document provides lesson materials on isosceles and equilateral triangles including:
- Key vocabulary terms like legs, vertex angle, and base of an isosceles triangle.
- The Isosceles Triangle Theorem and its converse.
- Properties and theorems regarding equilateral triangles.
- Examples proving triangles congruent using corresponding parts of congruent triangles (CPCTC).
- A lesson quiz to assess understanding of isosceles triangle properties and angle measures.
Parallelogram is a quadrilateral with two pairs of parallel sides.
There are 6 properties of parallelogram.
1. A diagonal of a parallelogram divides it into two congruent triangles.
2. Opposites sides of a parallelogram are congruent.
3. Opposite angles of a parallelogram are congruent.
4. Consecutive angles of a parallelogram are supplementary.
5. If one angle in a parallelogram is right, then all angles are right.
6. The diagonals of a parallelogram bisect each other.
Parallelogram is a quadrilateral with two pairs of parallel sides.
There are 6 properties of parallelogram.
1. A diagonal of a parallelogram divides it into two congruent triangles.
2. Opposites sides of a parallelogram are congruent.
3. Opposite angles of a parallelogram are congruent.
4. Consecutive angles of a parallelogram are supplementary.
5. If one angle in a parallelogram is right, then all angles are right.
6. The diagonals of a parallelogram bisect each other.
Use properties of similar triangles to find segment lengths
Apply proportionality and triangle angle bisector theorems
Use ratios to make indirect measurements
Use scale drawings to solve problems
The power point explains the concept of congruence in VII th standard .It explains the congruence of angles,vertices, triangles,quadrilaterals,and circle.
Use properties of similar triangles to find segment lengths
Apply proportionality and triangle angle bisector theorems
Use ratios to make indirect measurements
Use scale drawings to solve problems
The power point explains the concept of congruence in VII th standard .It explains the congruence of angles,vertices, triangles,quadrilaterals,and circle.
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1. Warm Up
1. Find each angle measure.
True or False. If false explain.
2. Every equilateral triangle is isosceles.
3. Every isosceles triangle is equilateral.
60°; 60°; 60°
True
False; an isosceles triangle can have only two
congruent sides.
Honors Geometry 1/14/15
3. Prove theorems about isosceles and
equilateral triangles.
Apply properties of isosceles and
equilateral triangles.
Objectives
4. legs of an isosceles triangle
vertex angle
base
base angles
Vocabulary
5. Recall that an isosceles triangle has at least
two congruent sides. The congruent sides
are called the legs. The vertex angle is the
angle formed by the legs. The side opposite
the vertex angle is called the base, and the
base angles are the two angles that have
the base as a side.
3 is the vertex angle.
1 and 2 are the base angles.
6.
7. The Isosceles Triangle Theorem is sometimes stated
as “Base angles of an isosceles triangle are
congruent.”
Reading Math
8. Example 1: Astronomy Application
The length of YX is 20 feet.
Explain why the length of YZ is the same.
Since YZX X, ∆XYZ is
isosceles by the Converse
of the Isosceles Triangle
Theorem.
The mYZX = 180 – 140,
so mYZX = 40°.
Thus YZ = YX = 20 ft.
9. The following corollary and its converse show the connection between equilateral
triangles and equiangular triangles.
10.
11. Example 3A: Using Properties of Equilateral Triangles
Find the value of x.
∆LKM is equilateral.
(2x + 32) = 60
The measure of each of an equiangular ∆ is 60°.
2x = 28 Subtract 32 both sides.
x = 14 Divide both sides by 2.
Equilateral ∆ equiangular ∆
12. Example 3B: Using Properties of Equilateral Triangles
Find the value of y.
∆NPO is equiangular.
Equiangular ∆ equilateral ∆
5y – 6 = 4y + 12
Definition of equilateral ∆.
y = 18
Subtract 4y and add 6 to both sides.
13. Check It Out! Example 3
Find the value of JL.
∆JKL is equiangular.
Equiangular ∆ equilateral ∆
4t – 8 = 2t + 1
Definition of equilateral ∆.
2t = 9
Subtract 4y and add 6 to both sides.
t = 4.5 Divide both sides by 2.
Thus JL = 2(4.5) + 1 = 10.
22. CPCTC is an abbreviation for the
phrase “Corresponding Parts of
Congruent Triangles are Congruent.”
It can be used as a justification in a
proof after you have proven two
triangles congruent.
24. SSS, SAS, ASA, AAS, and HL use corresponding parts
to prove triangles congruent. CPCTC uses congruent
triangles to prove corresponding parts congruent.
Remember!
25. Example 3: Using CPCTC in a Proof
Prove: MN || OP
Given: NO || MP, N P
26. 5. CPCTC5. NMO POM
6. Conv. Of Alt. Int. s Thm.
4. AAS4. ∆MNO ∆OPM
3. Reflex. Prop. of
2. Alt. Int. s Thm.2. NOM PMO
1. Given
ReasonsStatements
3. MO MO
6. MN || OP
1. N P; NO || MP
Example 3 Continued
27. Check It Out! Example 3
Prove: KL || MN
Given: J is the midpoint of KM and NL.
28. Check It Out! Example 3 Continued
5. CPCTC5. LKJ NMJ
6. Conv. Of Alt. Int. s
Thm.
4. SAS Steps 2, 34. ∆KJL ∆MJN
3. Vert. s Thm.3. KJL MJN
2. Def. of mdpt.
1. Given
ReasonsStatements
6. KL || MN
1. J is the midpoint of KM
and NL.
2. KJ MJ, NJ LJ
36. Lesson Quiz: Part I
Find each angle measure.
1. mR
2. mP
Find each value.
3. x 4. y
5. x
124°
28°
20 6
26°
37. Lesson Quiz: Part II
6. The vertex angle of an isosceles triangle measures (a +
15)°, and one of the base angles measures 7a°. Find a and
each angle measure.
a = 11; 26°; 77°; 77°
38. STW review all types of
triangles & triangle congruency
to prepare for Chapter test
