Table Of Contents Page One: Chapter Four Vocabulary Page Two: Identify and Classify Triangles by angles and sides. Page Three: Apply the Angle sum theorem and exterior angles sum theorem. Page Four: Solve Problems involving congruent triangles. Page Five: Prove congruent triangles using SSS and SAS Postulates. Page Six: Prove congruent triangles using ASA and AAS Postulates. Page Seven: Prove right triangles are congruent. Page Eight: Solve problems involving isosceles triangles & Solve Coordinate Proofs
Chapter 4 Vocabulary Acute Triangles- all angles equal less than 90 degrees Base Angles- Two angles formed by the base and one of the congruent sides. Congruent Triangles- Same Measure of sides and angles. Coordinate Proof- uses figures in a coordinate plane and algebra to prove a geometric concept. Corollary- a statement that can be easily proven using a theorem. Equiangular Triangle- An acute triangle with all angles congruent. Equilateral Triangle- All sides are congruent. Exterior Angle- formed by one side of a triangle and the extension of another side. Hypotenuse- the side of a triangle that the legs are attached to. Isosceles Triangles- at least two sides are congruent. Obtuse Triangles- One angle in the triangle is obtuse. Perpendicular Bisector- a bisector in a triangle perpendicular to the base forming two 90 degree angles. Remote Interior Angles- the interior angles not adjacent to the given exterior angle. Right Triangle- a triangle with one angle that equals 90 degrees Scalene- No sides of the triangle are congruent. Vertex Angle- Angle formed by the congruent sides.
Chapter Four identifying and classifying triangles by angles and sides.
iangles are triangles with one angle adding up to 90 degrees. The other two angles will add up to less than 90 degrees. Acute Triangles are triangles where all the angles add up to less than 90 degrees. Obtuse Triangles are triangles where one angle adds up to greater that 90 degrees.
Isosceles triangles are equal on two sides. Scalene Triangles are not equal on any sides. Equilateral Triangles has all congruent sides.
Chapter Four Apply the angle sum theorem and the exterior angle sum theorem. The sum of the measures of the angles of a triangle is 180 Angle Sum Theorem. Third Angle Theorem:
wo angles of a triangle are congruent to two angles of another triangle then the third proof angles are congruent. Exterior Angle Sum Theorem: The measure of an exterior angle is the sum of its two remote interior angles. EX:
e A and angle B should add together to get the Exterior Angle. The Exterior angle and its adjacent side should add together to equal 180 degrees. EX2:
A Corollary is the acute angles of a right triangle, which are complementary.
The corollaries in the image above would be angles one and two.
Chapter Four Solving problems involving Congruent Triangles. Congruent Triangles:
Triangles are congruent if and only if Their corresponding parts are congruent. In the photo above, we are given that Side NM and OL are congruent. We are also given that side NO is congruent to side ML. The polygon is bisected by a bisector of MO, which gives us two triangles. And dew to the Alt int angle postulate The two triangles are congruent. Example.
Chapter Four Prove congruent triangles using SAS and SSS Postulates.
When determining which postulate you must use you have to look at the sides and angles of the triangle. The SSS Postulate If three sides of a triangle are congruent to three sides Of another triangle the triangle’s are congruent by the SSS Postulate. Example (D) from above. Lets just say that. If ST is congruent to WV (S) TU is congruent to WX (S) And XV is congruent to US (S) Then the triangles is congruent by SSS Postulate. The SAS Postulate 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 by the SAS postulate. Example ( C ) from above. Lets just say that NO and RO are congruent (S) MO and PO are congruent (S) And the two included sides are congruent. (A) Then the Two triangles are congruent by using The SAS Postulate. Chapter Four Proving congruent angles by using ASA and AAS Postulates.
ASA Postulate If Two angles of a Triangle and the included side are congruent to two Angles and an included side of another triangle Then the triangles are congruent. Example (E) From above. Lets assume that The vertical angles are congruent ( which is always true) (A) Angle T and Angle N are congruent by Alt. Int. Angles(A) And sides TO & OE are congruent. (S) Then The triangles are congruent by the ASA Postulate. AAS Postulate If Two angles and their non-included sides are congruent To two angles and their non-included side of another triangle then The triangles are congruent by AAS Postulate. Example (A) From Above Lets Assume that CB & FE are congruent. (S) Angle A and Angles D are congruent(A) And angle C and Angle F are congruent. (A) Then the triangles are congruent by the AAS Postulate. Chapter Four Proving Right Triangles are congruent. There are four ways to tell if a right triangle is congruent. You can use the Hypotenuse-Leg Theorem, The Leg-Angle Theorem, The Leg-Leg Theorem, and The Hypotenuse- Angle theorem.
The Hypotenuse-Leg Theorem
In the image congruency between the two triangles by the HL Theorem. Both of the hypotenuse’s are congruent to each other. A leg is also congruent to another leg. Which brings out the Hypotenuse-Leg Theorem. The Leg-Angle Theorem
In the Image above (F) Demonstrates the LA Theorem. In the image one leg is congruent to the leg on the other triangle. Angle I is congruent to Angle E. Which means the angles are congruent by the LA Theorem.
The Leg-Leg Theorem
The image above shows the Leg-Leg Theorem. It shows how leg AB is congruent to leg ED And how BC is congruent to EF. Which shows the LL Theorem.
The Hypotenuse-Angle theorem
This image demonstrates the Hypotenuse-Angle Theorem, also known as the HA theorem. hen using the Ha theorem just look for congruent hypotenuse’s and a congruent angle other than the right angle.
Chapter Four Solve problems involving isosceles Triangles & coordinate proofs. Isosceles Triangles Have two congruent sides.
And the Angles adjacent from the sides are congruent.
The Isosceles Triangle theorem states that angles opposite of the legs In an isosceles triangle are congruent. (Base angles are congruent. ) Coordinate Proofs. STEPS 1) Use origin as a vertex of the triangle. 2) Place at least one side of the triangle on the Axis. 3) Keep Triangle in the first quadrant whenever possible. 4) use coordinates that make computations as simple as possible.