Angela Guerra
Geometry
Mr. Rice
6/7/15
Chapter 4 & 5 Study Guide
4-1 Classifying Triangles
By angles
Acute triangle- all of the angles are acute.
Equiangular triangle – acute tri. With all angles congruent
Obtuse triangle- one angle is obtuse.
Right triangle- one angle is right (90°)
By sides
Scalene triangle- no two sides are congruent.
Isosceles triangle- at least two sides are congruent
Equilateral triangle- all sides are congruent.
4-2 Angles of Triangles
Angle Sum Theorem- the sum of the angles of a triangle is 180°
Third Angel Theorem- If 2 angles of one tri. are congruent to 2 angles of a second
tri. then the third angles of the tri. are congruent.
Exterior Angle Theorem- measure of ext. angle of a tri. is equal to the sum of the
measures of the 2 remote int. angles.
(9x+16)+ (6x+15)=(19x+3)
15x+31=19x+3
28=4x
x=7
4-3 Congruent Triagles
Conguent trianlges- only if corresponding parts are congruent
CPCTC- “corresponding parts of congruent tri. are congruent”.
4-4 Proving Congruence ---- SSS, SAS
Side-Side-Side Postulate- if sides of one tri. are congruent to the sides of a second
tri, then the tri. are congruent.
Side-Angle-Side Postulate- if two sides and the included angle of one triangle are
congruent to two sides and the included angle of another triangle, then these two
triangles are congruent.
4-5 Proving Congruence ---- ASA, AAS
Angle-Side-Angle Congruence- If two angles and the included side of a triangle are
congruent to two angles and the included side of another triangle, then the two
triangles are congruent.
Angle-Angle-Side Congruence- If two pairs of angles of two triangles are equal in
measurement, and a pair of corresponding non-included sides are equal in length,
then the triangles are congruent.
parts of Isoc. Tri:
 leg(s)
 base
 vertex angle
 base angles
4-6 Isosceles Triangles
Base Angles Theorem-If two sides of a tri. are congruent, then the angles opposite
those sides are congruent.
Converse of the Base Angles Theorem-If two angles of a tri. are congruent, then
the sides opposite those angles are congruent.
4-7 Triangles and Coordinate Proof
Steps to Coordinate Proof
Given the coordinates of the triangle's vertices, to prove that a triangle is isosceles
• plot the 3 points(optional)
• use the distance formula to calculate the length of each side of the triangle.
If any 2 sides have equal side lengths, then the triangle is isosceles.
Practice Problem 1) The coordinates of triangle BCD are B(8,2), C(11,13) and
D(2,6) Using coordinate geometry, prove that triangle BCD is an isosceles triangle.
Calculate all 3 distances
The distances
Triangle BCD is isosceles because 2 of the sides have equal side length
congruent)
(ie are
5-1 Bisectors, Medians, and Altitudes
Median- the segment that connects a vertex of a triangle to a midpoint on one of
the triangle's sides. Therefore every triangle has 3 different medians--as you can see
from the pictures below. The medians, in this diagram, are the red segments. POC=
centroid
Altitude- a segment from the vertex of a triangle to the opposite side and it must be
perpendicular to that segment (called the base). POC= orthocenter
Perpendicular Bisectors - is a line segment that is both perpendicular to a side of
a triangle and passes through its midpoint. POC= curcumcenter
Angle Bisectors- is a line segment that bisects one of the vertex angles of a
triangle. POC= incenter
5-2 Inequalities and Triangles
Enterior Angle Inequality- the measure of an exterior angle of a triangle is greater
than either of its two non-adjacent interior angles.
Thereom- If one angle of a triangle is longer than another side, then the angle
opposite the longer side has a greater measure than the angel opposite the shorter
side. + Vise Versa
Example- Figure 2 shows a triangle with sides of different measures.
List the angles of this triangle in order from least to greatest.
Figure 2 List the angles of this triangle in increasing order.
Because 6 < 8 < 11, then m ∠ N < m ∠ M < m ∠ P.
5-3 Indirect Proof
Indirect Proof-is the same as proving by contradiction, which means that the
negation of a true statement is also true. Indirect proof is often used when the
given geometric statement is NOT true. Start the proof by assuming the statement
IS true.
In the accompanying diagram,
is not isosceles.
Prove that if altitude
is drawn, it will not bisect
STATEMENTS
1
.
2
.
3
.
4
.
5
.
6
.
7
.
1
.
(Remember to
assume the opposite of the PROVE.)
Assume
2
.
3
.
4
.
5
.
REASONS
Given
Assumption leading to a contradiction.
Bisector of a segment divides the
segment at its midpoint.
Midpoint divides a segment into two
congruent segments.
The altitude of a triangle is a line
segment extending from any vertex of a
triangle perpendicular to the line
containing the opposite side.
6 Perpendicular lines meet to form right
. angles.
7 All right angles are congruent.
.
8
.
9
.
1
0
.
1
1
.
1
2
.
8 Reflexive Property
.
9 SAS - If two sides and the included
. angle of one triangle are congruent to the
corresponding parts of a second triangle,
the two triangles are congruent.
1 CPCTC - Corresponding parts of
0 congruent triangles are congruent.
.
1 An isosceles triangle is a triangle with
1 two congruent sides.
.
1 Contradiction steps 1 and 11
2
.
5-4 The Triangle Inequality
Triangle Inequality- states that for any triangle, the sum of the lengths of any two
sides must be greater than or equal to the length of the remaining side.[1][2] If x, y,
and z are the lengths of the sides of the triangle, then the triangle inequality states
that
So in a triangle ABC, |AC| < |AB| + |BC|. (Also, |AB| < |AC| + |CB|; |BC| < |BA| + |AC|.)
5-5 Inequalities Involving two Triangles
SAS Inequality/Hinge Theorem- If two sides of a triangle are congruent to two
sides of another triangle and the included angle in one triangle has a greater
measure than the included angle in the other, then the third side of the first triangle
is longer than the third side of the second triangle.