VidaaraPlay
Door 9 · Grade 8

📐 Triangles

Scalene: all sides different | Isosceles: two sides equal | Equilateral: all sides equal

▶ Play Door 9

Classification of Triangles and Angle Properties

What is a Triangle? A triangle is a closed figure with three sides, three angles, and three vertices. It is the simplest polygon.

Classification by Sides:

TypeDescriptionExampleScalene triangleAll three sides are different lengthsSides: 3cm, 4cm, 5cmIsosceles triangleAt least two sides are equalSides: 5cm, 5cm, 4cmEquilateral triangleAll three sides are equalSides: 6cm, 6cm, 6cm
Classification by Angles:

TypeDescriptionExampleAcute triangleAll angles Angles: 50°, 60°, 70°Right triangleOne angle = 90°Angles: 30°, 60°, 90°Obtuse triangleOne angle > 90°Angles: 120°, 30°, 30°
Angle Sum Property: The sum of the three interior angles of any triangle is 180°.

\[ ∠ A + ∠ B + ∠ C = 180° \]
Exterior Angle Property: An exterior angle of a triangle is equal to the sum of the two opposite interior angles.

\[ Exterior angle = Sum of two remote interior angles \]

Worked example

Problem: Classify the triangle with sides 7 cm, 7 cm, and 5 cm.

  1. - Two sides are equal (7 cm and 7 cm)
  2. - Third side is different (5 cm)
  3. - This is an **isosceles triangle
  4. - **
  5. *Example 2:

Answer: ** Isosceles triangle

Congruence of Triangles

What is Congruence? Two triangles are congruent if they have the same shape and same size. All corresponding sides are equal and all corresponding angles are equal.

Notation: \(△ ABC △ PQR\) means:

• \(AB = PQ\), \(BC = QR\), \(CA = RP\)
• \(∠ A = ∠ P\), \(∠ B = ∠ Q\), \(∠ C = ∠ R\)

Conditions for Congruence (SSS, SAS, ASA, RHS):

ConditionFull FormExplanationSSSSide-Side-SideAll three sides of one triangle equal to all three sides of anotherSASSide-Angle-SideTwo sides and the included angle are equalASAAngle-Side-AngleTwo angles and the included side are equalAASAngle-Angle-SideTwo angles and a non-included side are equalRHSRight-Hypotenuse-SideRight triangle with hypotenuse and one side equal
Important: AAA (Angle-Angle-Angle) is NOT a congruence condition (triangles can be similar but not congruent).

Worked example

Problem: In \(△ ABC\) and \(△ DEF\), \(AB = DE\), \(BC = EF\), and \(AC = DF\). Are the triangles congruent? If yes, by which condition?

  1. - All three sides of one triangle equal all three sides of the other
  2. - This satisfies **SSS** (Side-Side-Side) condition
  3. - **
  4. *Example 2:
  5. In \(△ PQR\) and \(△ XYZ\), \(PQ = XY\), \(PR = XZ\), and \(∠ P = ∠ X = 90°\).

Answer: ** Yes, by SSS congruence

Medians, Altitudes, and Construction of Triangles

What is a Median? A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side. Every triangle has three medians, and they intersect at the centroid.

Properties of Medians:

• A median divides the triangle into two triangles of equal area
• The centroid divides each median in the ratio \(2:1\) (vertex to centroid : centroid to midpoint)

What is an Altitude? An altitude of a triangle is a perpendicular line segment from a vertex to the opposite side (or its extension). Every triangle has three altitudes, and they intersect at the orthocenter.

Medians vs Altitudes:

FeatureMedianAltitudeGoes fromVertex to opposite sideVertex to opposite sideSpecial propertyMeets at midpointPerpendicular to opposite sideIntersection pointCentroidOrthocenter
Construction of Triangles (Using Ruler and Compass):

Given conditionsConstruction stepsSSS…

Worked example

Problem: In \(△ ABC\), \(D\) is the midpoint of \(BC\). If \(AD\) is a median and \(AB = AC\), what type of triangle is it?

  1. - \(AB = AC\) means the triangle is isosceles
  2. - Median to the base of an isosceles triangle is also the altitude
  3. - So \(AD\) is perpendicular to \(BC\)
  4. - **
  5. *Example 2:

Answer: ** Isosceles triangle with \(AB = AC\)