4.1 and 4.2 Notes on Classifying Triangles and Angles Measures Name

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1 . and. Notes on Classifying Triangles and Angles Measures Name Polygon: a closed figure made up of segments that do not cross each other except at endpoints. Triangle: a three sided polygon Classifying triangles by sides: Equilateral Isosceles Scalene Classifying triangles by angles: Equiangular Right Acute Obtuse Example: Classify each of the following triangles by sides and angles.. 3. Isosceles Triangle Vertex Angle: angle formed by Base Angles: angles formed by the base and a leg (they are ) Legs: two sides Base: the side Example: Triangle RST is an isosceles triangle. Angle R is the vertex angle. RS=x+7, ST= x-, RT=3x-5. Find x, RS, ST, RT. Given triangle DAR with vertices D(,6), A(,-5), and R(-3,0), use the distance Formula to show if triangle DAR is isosceles.

2 The interior angles of a triangle are the three angles formed inside of a triangle at each of the three vertices. The Interior Sum Theorem for triangles states that the sum of the measures of the three interior angles of a triangle always equals In ABC, the measure of A 70, and. Find the measure of R in the m B 50. What is m C? figure below. T 3 R Suppose m G x 3, m H 6x,. In LPQ, m L y, m P 3y, and m I x 7 in triangle GHI. What and m Q y 6. What is the is the value of x? value of y? m G m H m I 80 x 3 6x x 7 80 x x 7 x.5 -- M N P S 5 In the figure to the left, note that side MP has been extended in order to form angle on the outside of the triangle at vertex P. Angle is known as an exterior angle. An exterior angle of a triangle is an angle that forms a linear pair with one of the interior angles of a triangle. Cover up MN if necessary to clearly see the linear pair formed. Since and are the two angles which make up a linear pair, then their measures must add to equal 80. Hence, the sum of the measures of an exterior angle of a triangle and the interior angle from the same vertex is 80.

3 E In the figure to the left, extend one of the sides to construct an exterior angle to DEF at vertex F. F D Label the angle with the number 5. Did everyone in the class construct the same exterior angle? What can be learned from this? What would be the appropriate three-letter name for the interior angle at vertex F? Label the angle with the number 6. Complete the statement: m Let's summarize the learning to this point using the figure to the right. m m m 3. 3 Also, m 3 m. Freeze! If both of these quantities above equal 80, then they must be equal to each other!!! Thus, m m m 3 m 3 m! However, since the quantities on both sides of the equation have m 3, it can be subtracted from both sides, leaving: m m m Look at the figure again to consider the implications of this statement. In a triangle, the two interior angles that are nonadjacent to the exterior angle are called remote interior angles. In the figure, and are the remote interior angles in relation to. In addition, as proven above in the notes, m m m. If all of this is put together, then the resulting theorem can be formed: 3 Exterior Angle Theorem: The measure of an exterior angle of a triangle is equal to the sum of the measures of the remote interior angles. 3

4 N Consider the figure to the left. P Suppose m M 7, and angle measures 7. What is the measure of angle N? M Well, consider that M and N are the remote interior angles to exterior angle. Thus, let's use the Exterior Angle Theorem from above. m M m N m 7 m N 7 m N Find the values of x and y in the figure to the right. 3 x y - Homework on Triangles and Angles. Triangle UVW is a triangle in which m V 0 and m W 0. What is m U?. A right triangle (that's important to this problem) has one interior angle that measures 0.5. Provide the measures of the other two interior angles of the triangle y 73 p x What are the values of x and y. What is the value of p in the in the figure above? figure above? Consider for Questions 5-6: An exterior angle of a triangle has a measure of What is the measure of the interior angle with the same vertex? 6. If one remote interior angle measures 59, what is the measure of the other remote interior angle?

5 M J x x An exterior angle has been drawn in the figure to the left. Use the figure for Questions What is the value of x? K L 8. What are the names of the two remote interior angles of the exterior angle drawn? 9. Complete the statement: m K m L m Do the two exterior angles which can be formed from the same vertex of a triangle have the same measure? Think carefully about the Exterior Angle Theorem. -. A triangle has interior angles which measure 89, 5, and 7. What is wrong with this statement? I. Consider the figure to the left. Which of the angles listed G below has the greatest measure? 70 A) F D) IGH B) FGH E)There is not enough information. F C) H H A triangle has an exterior angle. Its two remote interior angles measure 6 and. 3. What is the measure of the exterior angle?. What is the measure of the other interior angle? Find x and the length of each side if triangle ABC is an isosceles triangle with. 5

6 6. Find x and the length of each side if triangle FGH is an equilateral triangle. 7. Find the measure of the sides of triangle XYZ and classify each triangle by its sides. X(7,6), Y(5,), Z (9,) For each triangle, find x and the measure of each side. 8. Triangle FGH is an equilateral triangle with FG=3x-0, GH=x+5, and HF= x Triangle JKL is isosceles with, JK=x-, KL= x+5, and LJ= x-. 0. Triangle MNP is isosceles with MN is two less than five times x, NP is seven more than two times x, and PM is two more than three times x.. Triangle RST is equilateral. RS is three more than four times x, ST is seven more than two times x, and TR is more than five times x. 6