Triangles Continued…

This Week Monday – Angle Theorems Tuesday – Inequality Theorem & Midsegment Wednesday – Isosceles & Equilateral Triangles Thursday – Centroid, Medians, etc… lesson Honors find the centroid value Friday -- Quiz

Monday – DO NOWS Find the measure of the missing angle

Angle Sum Theorem: The sum of the measures of the angles of a triangle is 180 degrees. Find the measures of <1, <2 and <3…

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.

Example of Third Angle Theorem Find m<TSU, m<R, m<Q, and m<U

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

Example of Exterior Angle Theorem Find the m<1

Let’s try something harder

Exit Ticket - Monday What three theorems did we learn today? Which theorem would be used to solve the following problem? Solve the triangle for x.

DO NOW - Tuesday Classify the following triangles by their sides

Spaghetti Experiment… Now using a piece of spaghetti we are going to make triangles. Break the spaghetti into three unequal parts.  The parts can be any size that you want, but they cannot be the same size.  Now, I want you to make a triangle out of your spaghetti pieces. How many students were able to create a triangle with your three pieces of spaghetti? How many students were NOT able to create a triangle with your three pieces of spaghetti? Why do you think some made triangles and others didn’t?

Triangle Inequality Theorem In order for three lines to make a triangle, the sum of any two sides must be larger than the third side Examples: Do the following sides make a triangle? 4, 9 and 12 5, 6 and 11 2, 18 and 21 8, 14 and 20

Example: XYZ where XY=8 & XZ=14, what are the possible measures for YZ? Draw a Picture….

In triangle PQR, PQ = 7.2 and QR = 5.2 What lengths can PR be?

Mid-segment Theorem Mid-segment of a triangle: is a segment that joins the midpoints of two sides of a triangle such that it is parallel to the third side of the triangle. The Mid-segment theorem: The mid-segment of a triangle joins the midpoints of two sides of a triangle such that its length is half the length of the third side of the triangle

Example JH is congruent to HF

Example

Exit ticket - Tuesday List the two theorems that we learned today. How do these theorems differ from the ones we learned yesterday?

DO NOW - Wednesday Use your phone to find the following deginitions Vertex Angle: Base Angle: Draw a picture of an isosceles triangle and label the angles.

Isosceles Triangle Theorem and it’s converse Isosceles Triangle Theorem: If two sides of a triangle are congruent, then the angles opposite of those sides are congruent. Converse Isosceles Triangle Theorem: If two angles of a triangle are congruent, then the sides opposite those angles are congruent

Find the m<1, m<2 and x Example Find the m<1, m<2 and x

Example Given the following coordinates, is the triangle an isosceles? E(2, -1), B(0,1), D(2, 3) Hint: Distance Formula

Example Find the values of x and y

Exit Ticket - Wednesday Define Isosceles Triangle: Define Equilateral Triangle: Question… Are Equilateral Triangles also Isosceles Triangles? Things that make you say huh…

DO NOW - THURSDAY What is the measure of the interior angles in the triangle Why is m<4 = m<1? Why is m<3 = m<5? What do we know about <2, <4, and <5?

Folding Triangle Activity

Lines in a Triangle A perpendicular bisector of a triangle is a line, segment or ray that passes through the midpoint of the side and is perpendicular to that side. The point of concurrency of the perpendicular bisectors of a triangle is called the circumcenter An angle bisector divides the angle into congruent parts. The angle bisectors are concurrent and their point of concurrency is called the incenter of the triangle. A median of a triangle is a segment whose endpoints are a vertex of a triangle and the midpoint of the side opposite the vertex. The point of concurrency for the medians of a triangle is called a centroid The altitude of a triangle is a segment from a vertex to the line containing the opposite side and perpendicular to the line containing that side. The intersection point of the altitudes of a triangle is called the orthocenter

Exit Ticket Name the four types of lines in a triangle and their point of concurrency