Properties of Triangles A triangle is an enclosed 3 – sided figure.

Properties of Triangles A triangle is an enclosed 3 – sided figure. We will use the symbol ∆ to name triangles.

Properties of Triangles A triangle is an enclosed 3 – sided figure. We will use the symbol ∆ to name triangles. The points at which sides meet are called vertices and are named using capital letters. A B C

Properties of Triangles A triangle is an enclosed 3 – sided figure. We will use the symbol ∆ to name triangles. The points at which sides meet are called vertices and are named using capital letters. A This is now ∆ABC B C

Properties of Triangles A triangle is an enclosed 3 – sided figure. We will use the symbol ∆ to name triangles. The points at which sides meet are called vertices and are named using capital letters. A This is now ∆ABC B C A triangle also has three angles created by each pair of sides.

Properties of Triangles A triangle is an enclosed 3 – sided figure. We will use the symbol ∆ to name triangles. The points at which sides meet are called vertices and are named using capital letters. A This is now ∆ABC B C A triangle also has three angles created by each pair of sides. - sides AC and BC create ∠ACB

Properties of Triangles A triangle is an enclosed 3 – sided figure. We will use the symbol ∆ to name triangles. The points at which sides meet are called vertices and are named using capital letters. A This is now ∆ABC B C A triangle also has three angles created by each pair of sides. - sides AC and BC create ∠ACB - sides AB and BC create ∠ABC

Properties of Triangles A triangle is an enclosed 3 – sided figure. We will use the symbol ∆ to name triangles. The points at which sides meet are called vertices and are named using capital letters. A This is now ∆ABC B C A triangle also has three angles created by each pair of sides. - sides AC and BC create ∠ACB - sides AB and BC create ∠ABC - sides AB and AC create ∠BAC

Properties of Triangles A triangle is an enclosed 3 – sided figure. We will use the symbol ∆ to name triangles. The points at which sides meet are called vertices and are named using capital letters. A This is now ∆ABC The three angles in a triangle will ALWAYS add up to 𝟏𝟖𝟎° B C A triangle also has three angles created by each pair of sides. - sides AC and BC create ∠ACB - sides AB and BC create ∠ABC - sides AB and AC create ∠BAC

Properties of Triangles A triangle is an enclosed 3 – sided figure. We will use the symbol ∆ to name triangles. The points at which sides meet are called vertices and are named using capital letters. A This is now ∆ABC If ∠ACB =48° and ∠BAC =82°, find ∠ABC The three angles in a triangle will ALWAYS add up to 𝟏𝟖𝟎° B C A triangle also has three angles created by each pair of sides. - sides AC and BC create ∠ACB - sides AB and BC create ∠ABC - sides AB and AC create ∠BAC

Properties of Triangles A triangle is an enclosed 3 – sided figure. We will use the symbol ∆ to name triangles. The points at which sides meet are called vertices and are named using capital letters. A This is now ∆ABC If ∠ACB =48° and ∠BAC =82°, find ∠ABC ∠ACB + ∠BAC + ∠ABC =180° The three angles in a triangle will ALWAYS add up to 𝟏𝟖𝟎° B C A triangle also has three angles created by each pair of sides. - sides AC and BC create ∠ACB - sides AB and BC create ∠ABC - sides AB and AC create ∠BAC

Properties of Triangles A triangle is an enclosed 3 – sided figure. We will use the symbol ∆ to name triangles. The points at which sides meet are called vertices and are named using capital letters. A This is now ∆ABC If ∠ACB =48° and ∠BAC =82°, find ∠ABC ∠ACB + ∠BAC + ∠ABC =180° 48°+82°+ ∠ABC =180° The three angles in a triangle will ALWAYS add up to 𝟏𝟖𝟎° B C A triangle also has three angles created by each pair of sides. - sides AC and BC create ∠ACB - sides AB and BC create ∠ABC - sides AB and AC create ∠BAC

Properties of Triangles A triangle is an enclosed 3 – sided figure. We will use the symbol ∆ to name triangles. The points at which sides meet are called vertices and are named using capital letters. A This is now ∆ABC If ∠ACB =48° and ∠BAC =82°, find ∠ABC ∠ACB + ∠BAC + ∠ABC =180° 48°+82°+ ∠ABC =180° 130°+ ∠ABC =180° The three angles in a triangle will ALWAYS add up to 𝟏𝟖𝟎° B C A triangle also has three angles created by each pair of sides. - sides AC and BC create ∠ACB - sides AB and BC create ∠ABC - sides AB and AC create ∠BAC

Properties of Triangles A triangle is an enclosed 3 – sided figure. We will use the symbol ∆ to name triangles. The points at which sides meet are called vertices and are named using capital letters. A This is now ∆ABC If ∠ACB =48° and ∠BAC =82°, find ∠ABC ∠ACB + ∠BAC + ∠ABC =180° 48°+82°+ ∠ABC =180° 130°+ ∠ABC =180° ∠ABC =180° −130° The three angles in a triangle will ALWAYS add up to 𝟏𝟖𝟎° B C A triangle also has three angles created by each pair of sides. - sides AC and BC create ∠ACB - sides AB and BC create ∠ABC - sides AB and AC create ∠BAC

Properties of Triangles A triangle is an enclosed 3 – sided figure. We will use the symbol ∆ to name triangles. The points at which sides meet are called vertices and are named using capital letters. A This is now ∆ABC If ∠ACB =48° and ∠BAC =82°, find ∠ABC ∠ACB + ∠BAC + ∠ABC =180° 48°+82°+ ∠ABC =180° 130°+ ∠ABC =180° ∠ABC =180° −130° ∠ABC =50° The three angles in a triangle will ALWAYS add up to 𝟏𝟖𝟎° B C A triangle also has three angles created by each pair of sides. - sides AC and BC create ∠ACB - sides AB and BC create ∠ABC - sides AB and AC create ∠BAC

Properties of Triangles EXAMPLE : Find the missing angle in the given triangle. A ? 42° B C

Properties of Triangles EXAMPLE : Find the missing angle in the given triangle. Solution : ∠𝐴+ ∠𝐵+ ∠𝐶=180° A ? 42° B C

Properties of Triangles EXAMPLE : Find the missing angle in the given triangle. Solution : ∠𝐴+ ∠𝐵+ ∠𝐶=180° ∠𝐴+90°+42°=180° A ? 42° B C Shows a 90° angle

Properties of Triangles EXAMPLE : Find the missing angle in the given triangle. Solution : ∠𝐴+ ∠𝐵+ ∠𝐶=180° ∠𝐴+90°+42°=180° ∠𝐴+132°=180° A ? 42° B C

Properties of Triangles EXAMPLE : Find the missing angle in the given triangle. Solution : ∠𝐴+ ∠𝐵+ ∠𝐶=180° ∠𝐴+90°+42°=180° ∠𝐴+132°=180° ∠𝐴=180° −132° A ? 42° B C

Properties of Triangles EXAMPLE : Find the missing angle in the given triangle. Solution : ∠𝐴+ ∠𝐵+ ∠𝐶=180° ∠𝐴+90°+42°=180° ∠𝐴+132°=180° ∠𝐴=180° −132° ∠𝑨=𝟒𝟖° A ? 42° B C

Properties of Triangles A rule to check if 3 line segments create a triangle: The sum of the smallest two sides MUST be larger than the biggest side.

Properties of Triangles A rule to check if 3 line segments create a triangle: The sum of the smallest two sides MUST be larger than the biggest side. EXAMPLE : If AB = 6 , BC = 10, and AC = 15, do these sides create a triangle ?

Properties of Triangles A rule to check if 3 line segments create a triangle: The sum of the smallest two sides MUST be larger than the biggest side. EXAMPLE : If AB = 6 , BC = 10, and AC = 15, do these sides create a triangle ? SOLUTION : YES because 6 + 10 > 15

Properties of Triangles A rule to check if 3 line segments create a triangle: The sum of the smallest two sides MUST be larger than the biggest side. EXAMPLE : If AB = 6 , BC = 10, and AC = 15, do these sides create a triangle ? SOLUTION : YES because 6 + 10 > 15 EXAMPLE #2 : Do the following line segments create a triangle ? XY = 34 , XZ = 20 , YZ = 10

Properties of Triangles A rule to check if 3 line segments create a triangle: The sum of the smallest two sides MUST be larger than the biggest side. EXAMPLE : If AB = 6 , BC = 10, and AC = 15, do these sides create a triangle ? SOLUTION : YES because 6 + 10 > 15 EXAMPLE #2 : Do the following line segments create a triangle ? XY = 34 , XZ = 20 , YZ = 10 SOLUTION : NO because 20 + 10 < 34

Properties of Triangles Side and angle relationships : The larger the side, the larger the angle that creates that side. The larger the angle, the larger the side that contains that angle.

Properties of Triangles Side and angle relationships : The larger the side, the larger the angle that creates that side. The larger the angle, the larger the side that contains that angle. In the given triangle, we have 3 given angles. Order the sides from smallest to largest. X 79° 37° 64° Y Z

Properties of Triangles Side and angle relationships : The larger the side, the larger the angle that creates that side. The larger the angle, the larger the side that contains that angle. In the given triangle, we have 3 given angles. Order the sides from smallest to largest. X SOLUTION: Angle order = 37° , 64° , 79° 79° 37° 64° Y Z

Properties of Triangles Side and angle relationships : The larger the side, the larger the angle that creates that side. The larger the angle, the larger the side that contains that angle. In the given triangle, we have 3 given angles. Order the sides from smallest to largest. X SOLUTION: Angle order = 37° , 64° , 79° Sides order = XZ , XY , YZ 79° 37° 64° Y Z

Properties of Triangles Another EXAMPLE : In the given triangle, we have 3 given sides. Order the angles from smallest to largest. P 26 18 Q R 11

Properties of Triangles Another EXAMPLE : In the given triangle, we have 3 given sides. Order the angles from smallest to largest. P SOLUTION: Side order = 11 , 18 , 26 QR , PQ , PR 26 18 Q R 11

Properties of Triangles Another EXAMPLE : In the given triangle, we have 3 given sides. Order the angles from smallest to largest. P SOLUTION: Side order = 11 , 18 , 26 QR , PQ , PR Angle order = ∠𝑄𝑃𝑅 , ∠𝑄𝑅𝑃 , ∠𝑃𝑄𝑅 26 18 Q R 11

Properties of Triangles Some special triangles : Equilateral Triangle : All sides and angles are equal ** the angles all = 60° 60° 10 10 60° 60° 10

Properties of Triangles Some special triangles : Equilateral Triangle : All sides and angles are equal ** the angles all = 60° Isosceles Triangle : Two sides and angles are equal 60° 10 10 60° 60° 10 21 21 70° 70°

Properties of Triangles Some special triangles : Right Triangle : Contains a 90° angle