Chapter 4 Ms. Cuervo

Vocabulary: Congruent -Two figures that have the same size and shape. -Two triangles are congruent if and only if their vertices can be matched up so that the corresponding parts (angles and sides) of the triangle are congruent.

Some Ways to Prove Triangles Congruent Some Ways to Prove Triangles Congruent Lesson 2

Postulate 12 (SSS Postulate) If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.

Postulate 13 (SAS Postulate) If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.

Postulate 14 (ASA Postulate) If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.

Using Congruent Triangles Lesson 3

Example Given: AB and CD bisect each other at M. Prove: AD is parallel to BC A D B C M

StatementsReasons 1.AB and CD bisect each other at M. 2.M is the midpoint of AB and CD 1.AM ≅ MB; DM ≅ MC 4. <AMD ≅ <BMC 5. ∆AMD ≅ ∆BMC 6. <A ≅ <B 7. AB is parallel to BC 1.Given 2.Def. of bisector of a segment 3.Def. of midpoint 4.Vertical <‘s are ≅ 5.SAS Postulate 6.Corr. Parts of ≅ triangles are ≅ 7.If two lines are cut by a transversal and alt. interior angles are ≅, then the lines are parallel.

A Line and A Plane are Perpendicular If and only if they intersect and the line is perpendicular to all lines in the plane that pass through the point of intersection.

A Way to Prove Two Segments or Two Angles Congruent 1. Identify two triangles in which the two segments of angles are corresponding parts. 2. Prove that the triangles are congruent. 3. State that the two parts are congruent, using the reason: corresponding parts of ≅ triangles are ≅

The Isosceles Triangle Theorems Lesson 4

Vocabulary for Isosceles Triangles Legs-the two congruent sides Base-the third side of the triangle Vertex Angle Leg Base Angles BASE

Theorem 4-1 The Isosceles Triangle Theorem If two sides of a triangle are congruent, then the angles opposite those sides are congruent. Given: AB ≅ AC Prove: <B ≅ <C A BC D

Theorem 4-2 If two angles of a triangle are congruent, then the sides opposite those angles are congruent. Given: <B ≅ <C Prove: AB ≅ AC A BC D

Corollary 1 An equilateral triangle is also equiangular Corollary 2 An equilateral triangle has three 60 degree angles Corollary 3 The bisector of the vertex angle of an isosceles triangle is perpendicular to the base at its midpoint. Corollary 4 An equiangular triangle is also equilateral

Others Methods of Proving Triangles Congruent Lesson 5

Vocabulary Hypotenuse-In right triangles, the side opposite the right angle Legs-The other two sides

Theorem 4-3 (AAS Theorem) If two angles and a non- included side of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent. Given: ∆ABC and ∆DEF; <B ≅ <E; <C ≅ <F: AC ≅ DF Prove: ∆ABC ≅ ∆DEF

Theorem 4-4 HL Theorem If the hypotenuse and a leg of one right triangle are congruent to the corresponding parts of another right triangle, then the triangles are congruent Given: ∆ABC and ∆DEF; <C and <F are right <‘s AB ≅ DE BC ≅ EF Prove: ∆ABC ≅ ∆DEF

Summary of Ways to Prove Two Triangles Congruent All Triangles SSSSASASAAAS Right Triangles HL

Using More than One Pair of Congruent Triangles Lesson 6

Example Given : <1 <2; <5 Given : <1 ≅ <2; <5 ≅ <6 Prove: AC is perpendicular to BD A B C D

Statements Reasons 1.<1 ≅ <2; <5 ≅ <6 2.AC ≅ AC 3.∆ABC ≅ ∆ADC 4.AB ≅ AD 5.AO ≅ AO 6.∆ABO ≅ ∆ADO 7.<3 ≅ <4 8.AC is perpendicular to BD 1.Given 2.Reflexive Property 3.ASA Postulate 4.Corr. Parts of ≅ triangles are ≅ 5.Reflexive Property 6.SAS Postulate 7.Corr. Parts of ≅ triangles are ≅ 8.If two lines form ≅ adj <‘s, then the lines are perpendicular

Medians, Altitudes, and Perpendicular Bisectors Lesson 7

Vocabulary Median: the segment from a vertex to the midpoint of the opposite sides Altitude: the perpendicular segment from the vertex to the line that contains the opposite side.

Vocabulary Perpendicular Bisector: a line (or ray or segment) that is perpendicular to the segment at its midpoint. Distance from a Point to a Line (or Plane): the length of the perpendicular segment from the point to the line (or plane).

Theorem 4-5 If a point lies on the perpendicular bisector of a segment, then the point is equidistant from the endpoints of the segment. Given: Line l is the perpendicular bisector of BC; A is on line l Prove: AB = AC C A B X

Theorem 4-6 If a point is equidistant from the endpoints of a segment, then the point lies on the perpendicular bisector of the segment. Given: AB =AC Prove: A is on the perpendicular bisector of BC A CB X 12

Theorem 4-7 If a point lies on the bisector of an angle, then the point is equidistant from the sides of the angle Given: Ray BZ bisects <ABC P lies on ray BZ PX ⊥ BA PY ⊥ BC A B C Z X P Y

Theorem 4-8 If a point is equidistant from the sides of an angle, then the point lies on the bisector of the angle. Given: PX ⊥ BA; PY ⊥ BC PX=PY Prove: BP bisects <ABC A P CY X B