Download 6.2: Isosceles Triangles

6.2: Isosceles Triangles
Brinkman
Geometry
1. Define an Isosceles Triangle. A triangle that has (at least) two congruent sides.
2. Draw an Isosceles Triangle below to the best of your ability.
Mark the parts of congruency.
3. On your drawing above, identify the vertex, base and base angles.
4. What is true about the base angles of an Isosceles triangle? They’re congruent!
5. Define Isosceles Triangle Base Angles Theorem:
If a triangle has 2 ≅ sides then, the angles
opposite them are ≅.
6. Define Isosceles Triangle Symmetry Theorem:
The line containing the angle bisector of the
Vertex angle of an isosceles triangle is a
Line of symmetry for the triangle.
7. Define Isosceles Triangle Coincidence Theorem:
In an isosceles triangle, the angle bisector, the
Line of symmetry, and the perpendicular bisector
to the base are all the same line.
8. Draw a circle with center “w”. Put points “a” & “b” on the circle. Draw radii “wa” and “wb” . Explain
why the triangle you just created is isosceles.
Triangle is isosceles because by definition of a circle, AW=WB.
Therefore, (at least) two sides of the triangle are congruent, making
this triangle isosceles.
In the above picture you drew, in a circle, what is segment AB called?
A chord of circle W
9. Proof time!
Given: PQR is an Isosceles triangle with PQ = QR.
Prove: 𝑚∠1 = 𝑚∠3.
Conclusion
Justification
1. PQR is isosceles
With PQ = QR.
2. 𝒎∠𝟏 = 𝒎∠𝟐
Given
3. 𝒎∠𝟐 = 𝒎∠𝟑
Vertical Angle Theorem
4. 𝒎∠𝟏 = 𝒎∠𝟑
Transitive Property of =
Isosceles Triangle Base Angles Theorem
10. In Circle Y at the right, 𝑚∠𝑌 = 23. What is 𝑚∠𝑋?
180 – 23 = 157
157/2 = 78.5
11. Proof time!
Given: ∆𝑀𝐼𝐶 𝑖𝑠 𝑖𝑠𝑜𝑠𝑐𝑒𝑙𝑒𝑠 𝑤𝑖𝑡𝑕 𝑣𝑒𝑟𝑡𝑒𝑥 𝑎𝑛𝑔𝑙𝑒 𝑀;
∆𝐻𝐴𝐶 𝑖𝑠 𝑖𝑠𝑜𝑠𝑐𝑒𝑙𝑒𝑠 𝑤𝑖𝑡𝑕 𝑣𝑒𝑟𝑡𝑒𝑥 𝐻.
Prove: ∠𝐼 ≅ ∠𝐴
Conclusion
Justification
1. ∆𝑀𝐼𝐶 𝑖𝑠 𝑖𝑠𝑜𝑠𝑐𝑒𝑙𝑒𝑠
𝑤𝑖𝑡𝑕 𝑣𝑒𝑟𝑡𝑒𝑥 𝑎𝑛𝑔𝑙𝑒 𝑀;
2. ∆𝐻𝐴𝐶 𝑖𝑠 𝑖𝑠𝑜𝑠𝑐𝑒𝑙𝑒𝑠
𝑤𝑖𝑡𝑕 𝑣𝑒𝑟𝑡𝑒𝑥 𝐻.
3. ∠𝑀𝐼𝐶 ≅ ∠𝑀𝐶𝐼
Isosceles Triangle Base Angles Theorem
4. ∠𝐻𝐶𝐴 ≅ ∠𝐻𝐴𝐶
Isosceles Triangle Base Angles Theorem
5. ∠𝑀𝐶𝐼 ≅ ∠𝐻𝐶𝐴
Vertical Angle Theorem
6. ∠𝑀𝐼𝐶 ≅ ∠𝐻𝐴𝐶
Transitive Property of ≅
Given
Given
12. Draw an Equilateral Triangle and mark the parts of congruencies.
13. Define Equilateral Triangle Symmetry Theorem:
Every equilateral triangle has 3 symmetry lines, which are the bisectors of the angles.
14. What is the degree measure of each angle of an equilateral triangle? 60 degrees
Therefore an equilateral triangle is also called an equiangular triangle.
15. How many lines of symmetry does an isosceles triangle have? 1 line of symmetry
16. How many lines of symmetry does an equilateral triangle have? 3 lines of symmetry
17. Is an isosceles triangle equilateral? NO (only two sides congruent)
18. Is an equilateral triangle isosceles? YES (at least two sides are congruent)
Complete the remaining notes before starting the HW: May be done individually or with
partner.
A. Name the parts of the triangle given below:
A
B
D
C
∠𝐴____________________
𝐴𝐶 ____________________
∠𝐵____________________
𝐷 ____________________
∠𝐶____________________
𝐴𝐷____________________
𝐴𝐵 ____________________
𝐵𝐶 ____________________
Which angles are congruent in this triangle? _______________
What is 𝐴𝐷 called? _______________________________________
B. Name the parts in the figure given below.
B
∠𝐴____________________
𝐴𝐵 ____________________
∠𝐵____________________
𝐴𝐶 ____________________
∠𝐶____________________
𝐵𝐶 ____________________
A
Which angles are congruent in this triangle? _______________
If 𝑚∠𝐴 = 80o, find 𝑚∠𝐵 ________________
C
C. What type of triangle is ABC? __________________
find 𝑚∠𝐶 ________________
How many degrees is each angle? ______________
How many lines of symmetry does this triangle have? ________
Draw in lines of symmetry.
D. Find the indicated measures using the figure below and the given information.
In triangle ROT, 𝑚∠𝑂 = 132.
𝑚∠𝑅 = __________
𝑚∠𝑇 = __________
E. ∆𝐼𝑆𝑂 is isosceles with base 𝐼𝑂. If 𝑚∠𝐼 = (40 − 3𝑥) and 𝑚∠𝑂 = 5𝑥 − 8 , 𝑓𝑖𝑛𝑑 x, and the measure of each
angle.
X ___________ 𝑚∠𝐼 ___________
𝑚∠𝑂 _______ 𝑚∠𝑆 ___________