Vocabulary
• Acute Triangles- all angles are equal to less then 90 degrees • Base Angles- twp angles formed by the base and one of the congruent sides. • Congruent Triangles- triangles that are the same size and shape. • Coordinate proof- uses figures in the coordinate plane and algebra to prove geo concept. • Corollary- a statement that can be easily proven using a theorem. • Equiangular Triangle- an acute triangle in which all angles are congruent. • Equilateral Triangle- a triangle in which all of the sides are congruent. • Exterior angle- formed by one side of a triangle and the extension of another side. • Isosceles- a triangle in which at least two sides are congruent. • Obtuse Triangle- a triangle in which one angle is obtuse. • Remote interior angles- the interior angles not adjacent to a given exterior angle. •
Right Triangle- a triangle in which one angle equals 90 degrees.
• Scalene- a triangle in which no sides are congruent. • Vertex angle- an angle formed by the congruent sides.
Classifying Triangles Triangle- all of the angles measure less then 90 degrees.
Triangle- one of the angles measure more than 90 degrees.
Right Triangle- one angle measures 90 degrees.
Triangle- no two sides are congruent.
Isosceles Triangle- at least two sides are congruent.
Equilateral Triangles- all of the sides are congruent.
Angles of Triangles
ANGLE SUM THEOREM- the sum of the measures of the angles of a triangle is 180. For Example- measure of angle b + measure of angle a +
measure of angle c = 180 Third Angle Theorem- if two angles of one triangle are congruent to two angles of a second triangle, then the third angles of the triangles are
congruent. For example-
if angle C is congruent to angle F
and angle B is congruent to angle E, then angle A is congruent to angle D.
Exterior Angle Theorem- the measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles. For
example-
measure of angle RQS = measure of angle QPR +
measure of angle PRQ. Corollaries- The acute angles of a right triangle are complementary.
Measure of angle A + measure of angle C = 90
There can be at most one right or obtuse angle in a triangle.
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Congrutent Triangles Triangles are congruent if and only if their corresponding parts are congruent. Triangle ABC is congruent to triangle DEF, this is a congruence statement. Triangle ABC is congruent to triangle DEF Triangle ACB is congruent to triangle DFE Segment ED is congruent to segment BA Angle EDF is congruent to angle BAC Example- Triangle QRS is congruent to triangle GHJ QR = 5x- 11 GH = 2x+8 Find x and GH QR =GH 5x-11= 2x+8 -2x +11 -2x +11 3x= 19 X = 19/3
GH= 2(19/3) +8 38/3 + 24/3 GH= 62/3
Proving CongruenceSAS,SSS
Side-Side-Side PostulateSSS- If three sides of a triangle are congruent to three sides of another triangle then the triangles are congruent by SSS Example1. 2. 3. 4. 5.
Given T is the midpoint of segment SQ Prove- triangle SRT is congruent to triangle QRT
T is the midpoint of segment SQ Given Segment ST is congruent to segment TQ(S) Definition of midpoint Segment SR is congruent to segment QR(S) Given Segment RT is congruent to segment RT(S) Reflexive Triangle SRT is congruent to triangle QRT SSS
Side Angle Side postulate- SAS If two sides of a triangle are congruent to two corresponding sides of another triangle and the included angles are congruent then the triangles are congruent
ExampleGiven- segment RQ is parallel to segment TS and segment RQ is congruent to segment TS Prove- triangle QRT is congruent to triangle STR 1. Segment RT is congruent to segment RT Reflexive 2. Segment RT is parallel to segment Given 3. Angle 1 is congruent to angle 2 If lines Parallel then alt. int. angles are congruent 4. Segment RQ is congruent to segment TS Given 5. Triangle QRT is congruent to triangle STR SAS