Download Triangle Congruence Re

Name: ________________________________________ Date: __________________ Block: _________
Triangle Congruence Re-Visit
Section G.5: Properties of Triangle Lengths and Angle Measures
Triangle Basics:
• Know how to classify triangles
o By Angles: Acute, Obtuse, Right, Equiangular
o By Sides: Scalene, Isosceles, Equilateral
• Know the triangle theorems:
o Triangle Sum Theorem: the sum of the interior angles of a triangle is 180
o Exterior Angle Theorem: the exterior angle of a triangle is equal to the sum of the two
nonadjacent interior angles
o Third Angles Theorem: if 2 angles of one triangle are congruent to 2 angles of another
triangle, then the 3rd angles are also congruent
o Theorem to Order Angles: largest angle is across from the largest side.
o Theorem to Order Sides: largest side is across from the largest angle.
o Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle is
greater than the length of the third side (must be greater, cannot be equal or less).
o Hinge Angle Theorem: if you have 2 pair of congruent sides for 2 triangles, the
greatest “hinge” angle is across from the longest 3rd side.
o Hinge Angle Converse Theorem: if you have 2 pair of congruent sides for 2 triangles,
the longest 3rd side is across from the largest “hinge” angle.
• Know special triangles and their properties
o Isosceles:
Parts: Legs (2 congruent sides), base (3rd side), vertex angle (formed by legs),
base angles (angles across from legs)
Base Angles Theorem: isosceles triangles have congruent base angles
Converse of Base Angles Theorem: if 2 angles are congruent, the triangle is
isosceles
o Equilateral:
All 3 sides are congruent
Corollary to the Base Angles Theorem: equilateral triangles are equiangular
Corollary to the Converse of the Base Angles Theorem: equiangular triangles are
equilateral
• Know triangle segment/centers vocabulary/properties
o Midsegment: parallel to third side, half the length of third side
o Perpendicular bisector: intersects a triangle side perpendicularly at its midpoint;
center is the circumcenter
o Angle bisector: creates two congruent angles at a vertex of the triangle; center the
incenter
o Median: intersects the vertex of a triangle and the midpoint of the opposite side;
center is centroid; distance from centroid to vertex is 2/3 of entire segment length
o Altitude: called the height of the triangle; center is orthocenter
Name: ________________________________________ Date: __________________ Block: _________
Triangle Congruence Re
Re-Visit
Example Problems:
1. Can you form a triangle with
the given lengths? If no, use the
triangle inequality to explain
why not.
a) 2, 6, 19
b) 3, 5, 8
2. Find all possible lengths of
the 3rd side of the triangle with
given length
lengths (be sure to
include units):
a) 11ft, 23 ft
b) 10m, 6m
3. List the sides from smallest
to largest:
4. List the angles from smallest
to largest:
5. Find the value of x. Then
classify the triangle by its
angles:
6. Complete the statement with
<, >, or =.
MN _____ LK
7. Find the value of x:
8. DE is a midsegment of
ABC. Find x and y.
9. Find x:
10. Q is the centroid of XYZ.
a) Find XQ:
b) Find XN:
c) Find XM:
Name: ________________________________________ Date: __________________ Block: _________
Triangle Congruence Re
Re-Visit
Section G.6: Triangle Congruence
Proving Triangles are Congruent:
o Triangles can be proven congruent by: SSS post
postulate, SAS postulate
ulate, HL theorem
(special case of SSA that proves congruency)
congruency), ASA postulate, AAS theore
eorem.
o Triangles cannot be proven congruent by: AAA or SSA
o Once triangles are
re proven congruent, corresponding parts can be concluded congruent
by Corresponding Parts of Congruent
ongruent Triangles are Congruent
ongruent (CPCTC).
(CPCTC)
o Third Angles Theorem: if 2 angles of one triangle are congruent to 2 angles of another
triangle, then the 3rd angles are also congruent
o Base Angles Theorem and Converse
Converse:: Two sides of a triangle are congruent IFF the angles
opposite them are congruent.
o Corollaries to Base Angle Theorem and Converse
Converse:: A triangle is equilateral IFF it is
equiangular.
Example problems:
1. State the missing congruence
statement to prove ∆ABD ≅ ∆CDB
CDB by the
stated theorem/ postulate. Given: ∆ABC
is isosceles. ∠ A and ∠ C are base
angles.
a) Prove by: ASA.
b) Prove by: HL
3. Are the
triangles
congruent? If so,
what congruence
postulate /
theorem can be
used? If not, why
not?
2. ABC ≅ VTU.
Find m<T. Find AB.
4. Are the triangles
congruent? If so,
what congruence
postulate / theorem
can be used? If not,
why not?
5. Use the coordinates to determine if ABC
LMN. If not, why not?
A (-5, 1), B (-4, 4), C (-2,
2, 3), L (0, 2), M (1, 5), N (3, 4)
[Do not graph! Hint: Use the distance formula on all sides]
6. Write a proof:
7. Write a proof:
Given:
≅
≅
Given: ABC is isosceles;
Prove: ABD ≅
Prove:
≅
CBD
bisects <ABC.