Triangle Relationships Chapter 4

Objectives:  Classifying triangles and finding their angle measures.  Using the Distance Formula, the Pythagorean Theorem, and its converse.  Showing relationships between a triangle’s sides and angles.

Sections  4.1 Classifying Triangles  4.2 Angle Measures of Triangles  4.3 Isosceles and Equilateral Triangles  4.4 The Pythagorean Theorem and the Distance Formula  4.5 The Converse of the Pythagorean Theorem  4.6 Medians of a Triangle  4.7 Triangle Inequalities

Classifying Triangles Section 4.1

Objectives  Identify and classify triangles by angles  Identify and classify triangles by sides

Key Vocabulary  Acute Triangle  Equiangular Triangle  Obtuse Triangle  Right Triangle  Equilateral Triangle  Isosceles Triangle  Scalene Triangle  Vertex

Definition  Triangle: a figure formed by three segments joining three noncollinear points.  Two methods of classifying or naming triangles:  Angles  Sides  Triangle notation: ∆  Name triangle using three letters, therefore the above triangle is ∆ ABC.

Parts of a Triangle  A triangle is a 3-sided polygon  The sides of ∆ABC are AB, BC, and AC  A vertex of a triangle is a point that joins two sides of the triangle.  The vertices of ∆ABC are A, B, and C  Two sides sharing a common vertex are adjacent sides  The third side is called the opposite side  All sides can be adjacent or opposite (it just depends which vertex is being used) adjacent Side opposite  A C A B

Name the side that is opposite the angle. a. AA b. BB c. CC SOLUTION a. BC is the side that is opposite  A. b. AC is the side that is opposite  B. c. AB is the side that is opposite  C. Example 1:

Classifying Triangles by Angles  Four Classifications  Acute  Obtuse  Right  Equiangular  All triangles have at least two acute angles, the third angle is used to classify the triangle.

Classifying Triangles by Angles Obtuse Obtuse 1 angle is obtuse (measure > 90°) Right 1 angle is right (measure = 90°) One way to classify triangles is by their angles… Acute Acute all 3 angles are acute (measure < 90°) ( ) ) Equiangular All 3 congruent acute angles (measure < 90 ˚ and ≅ )

Definition: ACUTE Triangle a triangle in which all angles are acute. Classifying Triangles by Angles

Definition: OBTUSE Triangle a triangle in which one of the angles is an obtuse angle. Classifying Triangles by Angles

Definition: RIGHT Triangle a triangle in which one of the angles is a right angle. Classifying Triangles by Angles

Definition: EQUIANGULAR Triangle a triangle in which all angles are congruent. **EQUIANGULAR applies to any figure in which all angles are congruent** Classifying Triangles by Angles

 Each of the classifications (acute, obtuse, right, equiangular) is a distinct group and should not be combined.  A common mistake is to place triangles into more than one of the angle classifications.  Example: a right triangle cannot be classified as an acute triangle.

The triangular truss below is modeled for steel construction. Classify  JMN,  JKO, and  OLN as acute, equiangular, obtuse, or right. Example 2:

Answer:  JMN has one angle with measure greater than 90, so it is an obtuse triangle.  JKO has one angle with measure equal to 90, so it is a right triangle.  OLN is an acute triangle with all angles congruent, so it is an equiangular triangle. Example 2:

Classifying Triangles by Sides  Triangles can also be classified according to the number of congruent sides they have.  Three classifications:  Equilateral  Isosceles  Scalene  To indicate that sides of a triangle are congruent, an equal number of hash marks is drawn on the corresponding sides.

Classifying Triangles by Sides Isosceles Isosceles 2 congruent sides Scalene Scalene no congruent sides Another way to classify triangles is by their sides… Equilateral Equilateral 3 congruent sides

Definition: EQUILATERAL a triangle in which all sides are congruent. **EQUILATERAL applies to any figure in which all sides are congruent** Classifying Triangles by Sides

Definition: ISOSCELES a triangle in which at least 2 sides are congruent. Base Angles Vertex Angle Classifying Triangles by Sides

Definition: SCALENE a triangle in which no sides are congruent.

Classify the triangle by its sides. a. b. c. SOLUTION Because this triangle has 3 congruent sides, it is equilateral. a. Because this triangle has no congruent sides, it is scalene. b. Because this triangle has 2 congruent sides, it is isosceles. c. Example 3:

ANSWER equilateral Classify the triangle by its sides. 1. ANSWER isosceles ANSWER scalene Your Turn:

Name the side that is opposite the angle. a. AA b. BB c. CC SOLUTION a. BC is the side that is opposite  A. b. AC is the side that is opposite  B. c. AB is the side that is opposite  C. Example 4:

Answer:  UTX and  UVX are isosceles. Identify the isosceles triangles in the figure if Isosceles triangles have at least two sides congruent. Example 5:

Identify the scalene triangles in the figure if Answer:  VYX,  ZTX,  VZU,  YTU,  VWX,  ZUX, and  YXU are scalene. Scalene triangles have no congruent sides. Example 6:

Identify the indicated triangles in the figure. a. isosceles triangles b. scalene triangles Answer:  ABC,  EBC,  DEB,  DCE,  ADC,  ABD Answer:  ADE,  ABE Your Turn:

ALGEBRA Find d and the measure of each side of equilateral triangle KLM if and Since  KLM is equilateral, each side has the same length. So 5 = d Example 7:

Next, substitute to find the length of each side. Answer: For  KLM, and the measure of each side is 7. Example 7: KL = 7 LM = 7 KM = 7

Answer: ALGEBRA Find x and the measure of each side of equilateral triangle if and Your Turn:

Review: Classifications of Triangles by Angles 33 polygons triangles Right One 90 ˚ ∠ Equiangular All ∠s ≅ Acute All ∠s < 90˚ Triangle – 3 sides Polygon right acute equiangular Obtuse One ∠ > 90˚ obtuse

Review: Classifications of Triangles by Sides 34 polygons triangles Scalene No sides ≅ Equilateral 3 sides ≅ Isosceles 2 sides ≅ Triangle – 3 sides Polygon scalene isosceles equilateral

Joke Time  What did the pony say when he had a cold?  I’m just a little horse!  What is Beethoven doing in his grave?  De-composing  What do you call an arrogant household bug?  A cocky roach.

Assignment  Sec. 4.1 Pg : #1 – 29 odd, 30 – 36 all, 37 – 65 odd