Bellwork- MA.912.G.1.3

In the figure below, AB is parallel to DC. Which of the following statements about the figure must be true?

         

  A. m∠DAB + m∠ABC = 180° B. m∠DAB + m ∠ CDA = 180° C. ∠BAD is congruent to ∠ADC D. ∠ADC is congruent to ∠ABC

Highlands Park is located between two parallel streets, Walker Street and James Avenue. The park

faces Walker Street and is bordered by two brick walls that intersect James Avenue at point C, as shown below

What is the measure, in degrees, of ∠ACB, the angle formed by the park’s two brick walls?

5-3 BISECTORS IN TRIANGLES AND 5.4 MEDIANS AND ALTITUDESGeometry Chapter 5 Relationships within Triangles

Concurrency of Perpendicular Bisector Theorem

The perpendicular bisectors of the sides of a triangle intersect at a point called the circumcenter of the triangle, which is equidistant from the vertices of the triangle.

Point of Concurrency of a Perpendicular bisectors of a triangle

When three or more lines intersect at one point

Practice Example

RE=AE 3(x+3)=21 3x+9=21 3x=21-9→3x=12 X=4 AN=KN 4y-3=9→4y=9+3 4y=12 Y=3

Solution

Practice Example

4+x=2x-12

X=16

2x=x+3

X=3

Solve for x

Example#1

5x-4=x+6

X=10/4

Solve for x. Solution

Example#2

Find the circumcenter of DEFG with E(4, 4), F(4, 2), and G(8, 2).

(6,3)

Solve for x

Concurrency of Angle Bisectors Theorem

The angle bisectors of a triangle intersect at a point called the incenter of the triangle, which is equidistant from the sides of the triangle.

Point of concurrency of the angle bisectors of a triangle

Practice Example

Find the value of x?

The angle bisectors intersect at P.

The incenter P is equidistant from the sides, so SP = PT.

Therefore, x = 9. Note that , the

continuation of the angle bisector, is not the correct segment to use for the shortest distance from P to .

Examples

Find the value of x?

X=14 X=2

X=16

Median: A segment from a vertex to the midpoint of the opposite side

Median of a triangles

The medians of ABC are AM BL,and CX.

The centroid is point D.

The point of concurrency of the medians is called the centroid.

Centroids

Altitude of a triangle

Altitude: The perpendicular segment from a vertex to the line that contains the opposite side.

In an acute triangle all of the altitudes are all inside the triangle.

Altitudes of Right and Obtuse Triangle

In a right triangle, two of the altitudes are the legs of the triangle and the third is inside the triangle.

In an obtuse triangle, two of the altitudes are outside of the triangle.

Orthocenter

The orthocenter is the point of concurrency for the altitudes.

The altitudes of ∆QRS are AM, BL, and CX.

The orthocenter is point V.

Examples Determine whether is a median, an

altitude, or neither.Altitude

Altitude

Median

Neither

Example

Find the coordinates of the orthocenter of ∆ABC.

A(6, 10), B(2, 2), C(10, 2)

Real World Connections

Ticket Out and Homework

Sect. 5-3 pg. 319 #'s 4,7,11 Pg. 322 #'s 4-8,10 Sect. 5-4 Pg327-8 #’s 7-

10,14,15 Pg331-2#’s 7-9,12

What is true of the segments that connect the incenter and circumcenter to each side of the triangle?

Homework Ticket Out