Triangles and their properties Unit 3 Triangles and their properties

Lesson 3.1 Classifying Triangles Triangle Sum Theorem Exterior Angle Theorem

Classification of Triangles by Sides Name Equilateral Isosceles Scalene Looks Like Characteristics 3 congruent sides At least 2 congruent sides No Congruent Sides

Classification of Triangles by Angles Name Acute Equiangular Right Obtuse Looks Like Characteristics 3 acute angles 3 congruent angles 1 right angles 1 obtuse angle

Example 1 You must classify the triangle as specific as you possibly can. That means you must name Classification according to angles Classification according to sides In that order! Example Obtuse isosceles

More Examples 600 8.6 8.6 2.5 1280 2.5 600 600 260 260 8.6 4.5 Equiangular Equilateral Obtuse Isosceles

And more examples… Draw a sketch of the following triangles And more examples… Draw a sketch of the following triangles. Use proper symbol notation Obtuse Scalene Equilateral Right Equilateral impossible

Proving the Sum of a triangle’s Angles What do we know about the two green angles labeled X? What do we know about the two yellow angles labeled Y? What do we know about the three angles at the top of the triangle? (the X, Y and Z)

Proving the Sum of a triangle’s Angles X=X because of Alternate interior angles. What do we know about the two yellow angles labeled Y? What do we know about the three angles at the top of the triangle? (the X, Y and Z)

Proving the Sum of a triangle’s Angles X=X because of Alternate interior angles. Y=Y because of Alternate interior angles. What do we know about the three angles at the top of the triangle? (the X, Y and Z)

Proving the Sum of a triangle’s Angles X=X because of Alternate interior angles. Y=Y because of Alternate interior angles. X+Z+Y=180 because they form a straight line.

Proving the Sum of a triangle’s Angles The sum of the interior angles must also be equal to 1800 X=X because of Alternate interior angles. Y=Y because of Alternate interior angles. X+Z+Y=180 because they form a straight line.

Triangle Sum Theorem The sum of the interior angles of a triangle is 180.

Let’s try some…

Proving the Exterior Angle Theorem a+b+c=b+d Substitution a+c=d Subtraction (subtract b from both sides)

Exterior Angle Theorem

Practice (7x+1)+38=10x+9 7x+39=10x+9 39=3x+9 30=3x 10=x

Lesson 3.2 Inequalities in one triangle

Side/Angle Pairs in a Triangle B C Angle A and the side opposite it are a pair Angle B and the side opposite it are a pair Angle C and the side opposite it are a pair

The Inequalities in One Triangle If it is the longest side, then it is opposite largest angle measure. If it is the shortest side, then it is opposite the smallest angle measure. If it is the middle length side, then it is opposite the middle angle measure.

and their converse, too! If it is the largest angle measure, then it is opposite the longest side. If it is the smallest angle measure, then it is opposite the shortest side. If it is the middle angle measure, then it is opposite the middle length side.

Examples: Order the angles from smallest to largest.

More examples…Order the angles from smallest to largest V 12 5.8 W U 11

Let’s practice the converse now Let’s practice the converse now! Order the sides from shortest to longest.

Week’s Schedule Mon: Lesson 3.3 Tue: Lesson 3.4 Wed: MEAP Thu: Quiz/Lesson 3.5 Fri: Practice test Mon: Review Unit 3 Tue: Unit 3 Test

Lesson 3.3 Base Angles Theorem and its Converse

Let’s talk; What do we know about the following triangles?

Base Angles Theorem and Converse

Examples: solve for x and/or y 7 If the two angles are equal and the interior angles of a triangle have a sum of 180, what is the measure of the two angles? 75 3x = 45 y+7 = 45 30 x = 15 y = 38

One more example… solve for x and y Using the value of x, the measure of the two angles are each… 55 degrees Using the triangle sum theorem the last angle measure is… 70 degrees 3x-11 = 2x+11 2y = 70 x –11 = 11 y = 35 x = 22

Corollaries (a statement that is easily proven using the original theorem)

Prove what the angles in an equilateral triangle MUST always be. If all the sides are the same, equilateral, then all the angles must be the same, equiangular. If one of the angles is x, then all of the angles must also be x. x + x + x = 180 (triangle sum) 3x = 180 (combine like terms) x = 60 (DPOE)

Examples y 5x = 60 2x – 3 = 7 x = 12 2x = 10 x = 5

One more! All sides are equal. Pick any two and set them equal to each other. Then solve for x. 12x – 13 = 2x + 17 10x –13 = 17 10x = 30 x = 3

Lesson 3.4 Altitudes, Medians, and Perpendicular Bisectors of Triangles

Putting old terms together… Perpendicular: Two lines that intersect at a right angle. Bisector: A segment, ray, or line that divides a segment into two congruent parts. Perpendicular bisector (of a triangle): A segment, ray, or line that is perpendicular to a side of a triangle at the midpoint of the side.

Is segment BD a perpendicular bisector? Explain! No, it is nether perp. nor a bisector. Yes, it is perp. to segment AC and divides it into two congruent parts. No, it is a bisector but is not perp. No, segment BD is perp. to segment BC, but is not its bisector

New vocabulary terms! Median of a Triangle: A segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side. Altitude of a Triangle: The perpendicular segment from a vertex to the opposite side or to the line that contains the opposite side

Is segment BD a Median? Altitude? Explain! Neither, it is not a bisector and it is not perp. Both, it is a bisector and is perp. Median, it is a bisector of segment AC D Altitude, it is perp. to segment BC

Special notes about Perp. Bisectors and Medians: All perp. bisectors are also medians. Some medians are perp. bisectors. If it’s not a median, then it is not a perp. bisector.

Special notes about Perp. Bisectors and Altitudes: All perp. bisectors are also altitudes. Some altitudes are perp. bisectors. If it’s not a altitude, then it is not a perp. bisector.

Lesson 3.5 Perimeter and Area of Triangles

Review: Identify the altitude of the following triangles.

Area Formula of a Triangle b is the base h is the height HINT: The base and the height always meet at a right angle

Find the area of the triangles

Find the area. 8

Formula for the perimeter of a triangle. P=a+b+c a, b, and c are the three sides of the triangle. HINT: Perimeter is the sum of all three sides of a trianlge

Find the perimeter of the triangles

Find the perimeter 31 P=31+10+36 P=77