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1.5: Relationships Within Triangles

Created by: CK-12

Midsegments of a Triangle

Pacing: This lesson should take one class period

Goal: This lesson introduces students to the concept of midsegments and the properties they hold.

Extension: Have students find the other midsegments to the triangle of the introduction

Students will need to measure XY to find its midpoint, call it C, and create two new line segments: \overline{AC} and \overline{BC}. Encourage students to analyze the three midsegments. What do the segments form? What relationships can be seen? The midsegments form a second triangle. \overline{AB} is parallel to \overline{XY}, \overline{AC} is parallel to \overline{ZY}, and \overline{BC} is parallel to \overline{XZ}.

Discuss the proof of the first section of the Midsegment Theorem with your class. It may be helpful to begin with the figure above, and with each new justification, add it to the drawing.

Extension! Have your students take the paragraph proof of the Midsegment Theorem and rewrite it into 2-column form.

Additional Example: Find x, and the lengths of DE and BC.

Perpendicular Bisectors in Triangles

Pacing: This lesson should take one class period

Goal: Students are introduced to the concept of the circumcenter of a triangle. The process of finding the circumcenter uses perpendicular bisectors, which students can either draw using a protractor or construct using the method found in the Using Congruent Triangles lesson.

Additional Example: Construct a circle passing through these three points:

Angle Bisectors in Triangles

Pacing: This lesson should take one class period

Goal: Students will learn how to inscribe circles within triangles using the concept of angle bisectors. Students should be familiar with angle bisectors, as there were presented in chapter one.

Summary: To inscribe circles within triangles, the center of the circle is the intersection of angle bisectors. To circumscribe circles about triangles, the center of the circle is the intersection of perpendicular bisectors.

To construct the angle bisectors, repeat the In-class Activity found in Using Congruent Triangles lesson.

Additional Example: Locate the circle inscribed within the following triangle.

Medians in Triangles

Pacing: This lesson should take one class period

Goal: This lesson introduces the centroid of a triangle. By now, students should be familiar with the three main intersection points regarding triangles: the circumcenter, the incenter, and the centroid.

History Connection! In addition to an infamous dictator, it appears Napoleon Bonaparte was an excellent mathematician. He was the top mathematics student in his school, taking algebra, trigonometry, and conics. His favorite class, however, was geometry. After graduation, Bonaparte interviewed for a position in the Paris Military School and was accepted due to his mathematical ability. Bonaparte completed the curriculum in one year (it took average students two or three years to complete) and was appointed to the mathematics section of the French National Institute.

During his reign, Bonaparte appointed such men as Gaspard Monge, Joseph Fourier, and Pierre Laplace to recruit teachers and reform the curriculum to emphasize mathematics. Napoleon’s Theorem is named as such because, while Napoleon was not the first person to discover it, he supposedly found it independently.  

Altitudes in Triangles

Pacing: This lesson should take one class period

Goal: Students will learn how to construct an altitude and how this auxiliary line differs from the median. Altitudes are important in such geometrical concepts as area and volume.

Guided Discovery Questions! What is the difference between a median and an altitude? Is a median always an angle bisector? Can the perpendicular bisector be a median?

Review Question: Using the diagram below, ask students to label the following auxiliary items: median, circumcenter, incenter, orthocenter, altitude, perpendicular bisector

Inequalities in Triangles

Pacing: This lesson should take one class period

Goal: The purpose of this lesson is to familiarize students with the angle inequality theorems and the Triangle Inequality Theorem. The lesson further extends the concepts of perpendicular lines and triangles to deduce the shortest path between a point and a line is its perpendicular, thus leading to parallel lines.

If you have not used the in-class activity found in chapter 1, Classifying Triangles lesson, include it here. Otherwise, reintroduce the concept in this lesson.

Additional Example: Jerry is across the street in the following diagram. Draw the path she should travel to minimize the distance across the street.

Inequalities in Two Triangles

Pacing: This lesson should take one class period

Goal: The purpose of this lesson is to utilize the concept of inequality to determine corresponding angle measures and side lengths of a triangle.

Review! Be sure your students can solve inequalities. Try these as a warm-up or brief review.

  1. 8x - 4 + x > -76 \ \ \ x > -8
  2. -3(4x - 1) \ge 15 \ \ \ x  \le -1
  3. 8y - 33 > -1 \ \ \  y > 4

Additional Examples:

List the sides of each triangle in order from shortest to longest.

1. \triangle ABC with m \angle A = 90, m \angle B =40, m \angle C = 50.\ \ AC, AB, BC

2. \triangle XYZ with m \angle X = 51, m \angle Y = 59, m \angle Z = 70.\ \ YZ, XZ,  XY

List the angles of the triangle in order from largest to smallest.

3. \triangle ABC where AB = 10, BC = 3, and CA = 9. \ \ \angle C, \angle B, \angle A,

Indirect Proof

Pacing: This lesson should take one class period

Goal: Students have seen several theorems proven using the indirect proof. Indirect proof is an invaluable resource to students attempting to prove theorems or postulates.

Indirect proofs typically have four sentences that can be summarized by the following acronym: ATBT.

A – Assume (the opposite of what you’re trying to prove. Essentially, this is the negation of the conclusion of the conditional)

T – Then (by doing some mathematics or using reasoning, a conclusion can be made)

B – But (here lies the contradiction. The conclusion you made in the previous sentences defies a definition or previously proved theorem)

T – Therefore (your original conditional must be true)

Indirect proofs are usually used when the word cannot appears in the proof.

Example: Prove a triangle cannot have two right angles.

Assume a triangle can have two right angles. Then, using the Triangle Sum Theorem, 90 + 90 + x = 180\;\mathrm{degrees.} Using the angle addition property and the addition property of equality, x = 0 (here lies your contradiction). But a triangle must have three angles greater than zero degrees. Therefore, a triangle cannot have two right angles.

Additional Example: Prove \sqrt{15} \neq 4.

Assume \sqrt{15} = 4. Then, by squaring both sides, 15 = 16. But 15 \neq 16. Therefore, \sqrt{15} \neq 4

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CK.MAT.ENG.TE.1.Geometry.1.5

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