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Triangle Relationships

Prove theorems about the sum of angles, base angles of isosceles triangles, and exterior and interior angles.

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Applications of Triangle Theorems

Find a piece of cardstock or thick paper. Use a ruler and pencil to draw a fairly large random triangle on the paper. Use your ruler to help you to construct the centroid of the triangle. Carefully cut out the triangle and try to balance it on the tip of your pencil. Where is the balancing point?

Applications of Triangle Theorems

There are nine theorems related to triangles that are helpful to know.

  1. The sum of the measures of the interior angles of a triangle is \begin{align*}180^\circ\end{align*}.
  2. The base angles of an isosceles triangle are congruent.
  3. If a triangle has two congruent angles then it is isosceles. (Note that this is the converse of #2)
  4. The measure of an exterior angle of a triangle is equal to the sum of the measures of the remote interior angles.
  5. The segment connecting two midpoints of a triangle is both parallel to and one half the length of the third side of the triangle.
  6. The three medians of a triangle meet at a point called the centroid. The centroid divides each median in a 2:1 ratio with the larger segment being the one from the vertex to the centroid.
  7. The three angle bisectors of a triangle meet at a point called the incenter.
  8. The three altitudes of a triangle meet at a point called the orthocenter.
  9. The three perpendicular bisectors of a triangle meet at a point called the circumcenter.

Solve the following problems

Point \begin{align*}P\end{align*} is the centroid of \begin{align*}\Delta ABC\end{align*}. Find the length of \begin{align*}\overline{AP}\end{align*}.

Because \begin{align*}P\end{align*} is the centroid, it divides each median in a 2:1 ratio, where the length of the segment from the vertex to the centroid is the longer segment. This means that \begin{align*}\overline{AP}\end{align*} is twice the length of the segment that is marked as \begin{align*}2 \ in\end{align*}. Therefore, \begin{align*}AP=4\end{align*}.

Solve for \begin{align*}x\end{align*}.

The sum of the measures of the angles is \begin{align*}180^\circ\end{align*}. Set up an equation and solve for \begin{align*}x\end{align*}.

\begin{align*}3x+15+3x-5+2x+2 &=180\\ 8x+12 &=180\\ 8x &=168\\ x &=21 \end{align*}

Prove that two of the medians of an isosceles triangle are congruent.

You can complete this proof using the triangle below. Your goal is to prove that \begin{align*}\Delta DBC\cong \Delta ECB\end{align*} and then show that the medians are congruent because they are corresponding parts of the triangles.

Consider isosceles \begin{align*}\Delta ABC\end{align*} with \begin{align*}\overline{AB}\cong \overline{AC}\end{align*}. Midpoints \begin{align*}D\end{align*} and \begin{align*}E\end{align*} divide \begin{align*}\overline{AB}\end{align*} and \begin{align*}\overline{AC}\end{align*} respectively, so \begin{align*}\overline{AD}\cong\overline{DB}\cong\overline{AE}\cong\overline{EC}\end{align*}. Because it is an isosceles triangle, \begin{align*}\angle ABC\cong \angle ACB\end{align*}. \begin{align*}\overline{BC}\cong \overline{BC}\end{align*} because any segment is congruent to itself. Therefore, \begin{align*}\Delta DBC\cong \Delta ECB\end{align*} by \begin{align*}SAS\cong\end{align*}. \begin{align*}\overline{DC}\cong \overline{EB}\end{align*} because corresponding parts of congruent triangles are congruent.

Examples

Example 1

Earlier, you were asked where is the balancing point. 

You should find that the centroid is the balancing point of the triangle. This means that the centroid is the center of gravity for the triangle when constructed in real life.

Example 2

Point \begin{align*}P\end{align*} is the centroid of \begin{align*}\Delta ABC\end{align*} and \begin{align*}BD=18\end{align*}. Find the length of \begin{align*}\overline{BP}\end{align*}.

The centroid divides the median in a 2:1 ratio where \begin{align*}\overline{BP}\end{align*} is the longer segment.

\begin{align*}2x+1x &=18 \\ x &=6\\ 2x &=12\end{align*}

The two segments have lengths of 6 and 12, so \begin{align*}BP=12\end{align*}.

Example 3

The three altitudes of a triangle meet at a point called the ____________.

orthocenter

Example 4

Solve for \begin{align*}x\end{align*}.

The measure of the exterior angle is equal to the sum of the measures of the remote interior angles. Set up an equation to solve for \begin{align*}x\end{align*}.

\begin{align*}11x-15 &=4x-2+6x+2\\ 11x-15 &=10x\\ x &=15 \end{align*}

Review

1. Point \begin{align*}P\end{align*} is the centroid of \begin{align*}\Delta ABC\end{align*} and \begin{align*}CD=12\end{align*}. Find the lengths of \begin{align*}\overline{CP}\end{align*} and \begin{align*}\overline{DP}\end{align*}.

2. What are the four points of concurrency for a triangle?

3. The three medians of a triangle meet at a point called the ____________.

4. The three angle bisectors of a triangle meet at a point called the _____________.

5. The three perpendicular bisectors of a triangle meet at a point called the _____________.

6. Investigate which points of concurrency are always inside a triangle and which points of concurrency are sometimes outside a triangle (use geometry software to help). What did you find out?

7. Explore the points of concurrency for an equilateral triangle (use geometry software to help). What do you notice?

Solve for \begin{align*}x\end{align*}.

8.

9.

10.

11. Solve for \begin{align*}x\end{align*} and \begin{align*}y\end{align*}.

12. Solve for \begin{align*}x\end{align*}.

In the triangle below, \begin{align*}BC=x\end{align*}, \begin{align*}AE=x-2\end{align*}\begin{align*}D\end{align*} is the midpoint of \begin{align*}\overline{AB}\end{align*}, and \begin{align*}E\end{align*} is the midpoint of \begin{align*}\overline{AC}\end{align*}, and the perimeter of the triangle is 42.

13. Solve for \begin{align*}x\end{align*}.

14. Find \begin{align*}DE\end{align*}, \begin{align*}BC\end{align*}, \begin{align*}AE\end{align*}, and \begin{align*}AD\end{align*}.

15. A kite has diagonals with lengths 4 and 6. An inner quadrilateral is formed by joining the midpoints of each of the four sides of the kite. What is the perimeter of this inner quadrilateral?

Answers for Review Problems

To see the Review answers, open this PDF file and look for section 4.6. 

Vocabulary

altitude

An altitude of a triangle is a line segment from a vertex and is perpendicular to the opposite side. It is also called the height of a triangle.

angle bisector

An angle bisector is a ray that splits an angle into two congruent, smaller angles.

Median

The median of a triangle is the line segment that connects a vertex to the opposite side's midpoint.

perpendicular bisector

A perpendicular bisector of a line segment passes through the midpoint of the line segment and intersects the line segment at 90^\circ.

theorem

A theorem is a statement that can be proven true using postulates, definitions, and other theorems that have already been proven.

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