Chapter 4: Congruent Triangles

4-1 Congruent Figures Congruent- when two figures have the same size and shape

4-1 Continued Congruent triangles- 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 Their corresponding angles are congruent because congruent triangles have the same shape. Their corresponding sides are congruent because congruent triangles have the same size.

4-1 Continued Congruent parts of triangles are marked alike. Congruent triangles must be named in the same order of congruency. SUN RAY

4-1 Continued When justifying statements by use of the definition of congruent triangles, use this wording: Corresponding parts of congruent triangles are congruent, which is written: Corr. Parts of s are .

4-1 Continued Congruent polygons- two polygons are congruent if and only if their vertices can be matched up so that their corresponding parts are congruent ABFGH BCDEF

4-2 Some Ways to Prove Triangles Congruent Proving triangles congruent with only three corresponding parts. 1. Side Side Side Postulate (SSS)- if three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent

4-2 Continued Side Angle Side Postulate (SAS)- 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

4-2 Continued Angle Side Angle Postulate (ASA)- 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

Proof of ASA Postulate Statement Reason Given: Prove: Given E is the midpoint of 1. E is the midpoint of Given Definition of a midpoint If two lines are perpendicular then they form congruent adjacent angles. 5. Reflexive property of congruence 6. SAS postulate 2. 3. 4. 5. 6.

4-3 Using Congruent Triangles Learning how to extract information on segments or angles once it is shown that they are corresponding parts of congruent triangles…

4-3 Continued Statement Reason Given: AB and CD bisect each other at M 1. and Given 2. Definition of a bisector of a segment 3. Definition of a midpoint 4. Vertical angles are congruent 5. SAS Postulate 6. Corresponding parts of congruent triangles are congruent 7. If two lines are cut by a transversal and alternate interior angles are congruent, then the lines are parallel. Given: AB and CD bisect each other at M Prove: AD BC bisect each other at M 2. M is the midpoint of and of ll 3. ; 4. 5. 6. 7. ll

4-3 Continued 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.

4-3 Continued Statement Reason Given: PO plane X; AO BO Prove: PA 1. plane X 1. Given 2. Definition of a line perpendicular to a plane. 3. Definition of perpendicular lines 4. Defintion of congruent angles 5. Given 6. Reflexive Property 7. SAS postulate 8. Corresponding parts of congruent angles are congruent. 2. ; Given: PO plane X; AO BO Prove: PA 3. m = 90; m = 90 4. 5. 6. 7. 8.

4-3 Continued To prove two segments or two angles are congruent: 1.) Identify two triangles in which the two segments or angles are corresponding parts. 2.) Prove that the triangles are congruent. 3.) State that the two parts are congruent, using this reason Corr. Parts of s are .

4-4 The Isosceles Triangle Theorems Legs- the congruent sides of a triangle Base- the non-congruent side of a triangle Base angles- the angles at the base of the triangle Vertex angle- the angle opposite the base of the isosceles triangle Vertex angle Leg Leg Base angles Base

4-4 Continued The Isosceles Triangle Theorem- if two sides of a triangle are congruent, then the angles opposite those sides are congruent

4-4 Continued 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

4-4 Continued Theorem 4-2 If two angles of a triangle are congruent, then the sides opposite those angles are congruent. Corollary- an equilateral triangle is also equilateral * Theorem 4-2 is the converse of Theorem 4-1, and the corollary of Theorem 4-2 is the converse of Corollary 1 of Theorem 4-1.

4-5 Other Methods of Proving Triangles Congruent Angle Angle Side Theorem (AAS)- 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

4-5 Continued Hypotenuse- the side opposite the right angle in a right triangle Legs- the other two sides of the triangle hypotenuse leg leg

4-5 Continued Hypotenuse Leg 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

4-5 Continued Leg-Leg Method- if two legs of one right triangle are congruent to the two legs of another right triangle, then the triangles are congruent Hypotenuse-Acute Angle Method- if the hypotenuse and an acute angle of one right triangle are congruent to the hypotenuse and an acute angle of another right triangle, then the triangles are congruent Leg-Acute Angle Method- If a leg and an acute angle of one right triangle are congruent of the corresponding parts in another right triangle, then the triangles are congruent.

4-6 Using More than One Pair of Congruent Triangles 1. 1. 4-6 Using More than One Pair of Congruent Triangles Statement Reason Given: Prove: Given Reflexive property ASA postulate Corresponding parts of congruent angles are congruent. 5. Reflexive property 6. SAS postulate (1, 4, 5) 7. Corresponding parts of 8. If two lines form congruent adjacent angles, then the lines are perpendicular. 1. ; 2. 3. 4. 5. 6. 7. 8.

4-7 Medians, Altitudes, and Perpendicular Bisectors Median- a segment from a vertex to the midpoint of the opposite side in a triangle

4-7 Continued Altitude- the perpendicular segment from a vertex to a line that contains the opposite side In an acute triangle, the three altitudes are all inside the right triangle.

4-7 Continued In a right triangle, two of the altitudes are parts of the triangle. They are the legs of the right triangle. The third altitude is inside the triangle. In an obtuse triangle, two of the altitudes are outside the triangle.

4-7 Continued Perpendicular bisector- a line (or ray or segment) that is perpendicular to the segment at its midpoint

4-7 Continued 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.

4-7 Continued 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. *Theorem 4-6 is the converse of Theorem 4-5.

4-7 Continued The distance from a point to a line (or plane) is defined to be the length of the perpendicular segment from the point to the line (or plane).

4-7 Continued Theorem 4-7 If a point lies on the bisector of an angle, then the point is equidistant from the sides of the angle.

4-7 Continued Theorem 4-8 If a point is equidistant from the sides of an angle, then the point lies on the bisector of the angle. * Theorem 4-8 is the converse of Theorem 4-7.

The End (Thank God!)