Day 31 – Interior angles of a triangle

Introduction A triangle is a common three-sided plane figure in geometry. It is general knowledge that a triangle has three sides. Consequently, it has three interior angles positioned between any two of its edges. These three angles are enclosed in the triangle. In this lesson, we are going to show that the three angles inside a triangle sum to 180°.

Vocabulary Triangle A plane figure bounded by three line segments to form its edges and three vertices formed between two adjacent edges. Interior angles Angles inside a plane figure formed between two adjacent edges.

Interior angles of a triangle A triangle has three interior angles Interior angles of a triangle A triangle has three interior angles. They add up to 180°. We are going to show that these three interior angles add up to 180° using the properties of angles formed between a pair of parallel lines when intersected by a transversal line.

∠𝑎, ∠𝑏 and ∠𝑐 are interior angles of the triangle PQR shown below ∠𝑎, ∠𝑏 and ∠𝑐 are interior angles of the triangle PQR shown below. We want to show that ∠𝑎+∠𝑏+∠𝑐=180° a b c P Q R

If we draw line ST parallel to QR through point P and then extend line QR in both directions, different types of angles are formed between the two parallel lines as shown below. a b c P Q R S T

∠QPS and ∠PQR are alternate interior angles formed by the transversal line through points P and Q; therefore the two angles are congruent. This means that: ∠QPS=∠PQR but ∠𝐏𝐐𝐑=∠𝒂, hence ∠𝐐𝐏𝐒=∠𝒂 Similarly, ∠RPT and ∠PRQ are alternate interior angles formed by the transversal line through points P and R; therefore the two angles are also congruent. ∠RPT=∠PRQ but ∠𝐏𝐑𝐐=∠𝒄, hence ∠𝐑𝐏𝐓=∠𝒄

Now, let us indicate the new angles on the figure as shown below Now, let us indicate the new angles on the figure as shown below. Clearly, ST is a straight line and angles on a straight line add up to 180°. This means that ∠𝑎+∠𝑏+∠𝑐=180° a b c P Q R S T

We have seen that ∠𝑎, ∠𝑏 and ∠𝑐 are the interior angles of the triangle. We have therefore shown that; ∠𝒂+∠𝒃+∠𝒄=𝟏𝟖𝟎° This confirms that the sum of interior angles of a triangle add up to 180°.

Example In the figure below, PR∥ST and the two transversal lines intersect the pair of parallel lines to form triangle ABC as shown below. x y z A S T P R B C 63° 72°

(a) Find the measure of ∠𝑥 (b) Find the measure of ∠𝑦 (c) Find the measure of ∠𝑧 (d) Find the sum ∠𝑥+∠𝑦+∠𝑧. What information can be deduced from that sum with respect to the triangle? Solution (a) PR is a straight line and angles on a straight line add up to 180°. 72°+∠𝑥+63°=180° ∠𝒙=𝟒𝟓° (b) ∠𝑦 is alternate to 72° and alternate angles are equal. ∠𝒚=𝟕𝟐°

(c)∠z is alternate to 63° and alternate angles are equal (c)∠z is alternate to 63° and alternate angles are equal. ∠𝒛=𝟔𝟑° (d) ∠𝑥+∠𝑦+∠𝑧=45°+72°+63°=180°. The interior angles of the triangle add up to 180°.

homework In the figure below, KL∥MN and the two transversal lines intersect the pair of parallel lines to form triangle XYZ as shown below. a b c X M N K L Y Z 58° 49°

(a) Find the measure of ∠𝑎 (b) Find the measure of ∠𝑏 (c) Find the measure of ∠𝑐 (d) Find the sum ∠𝑎+∠𝑏+∠𝑐. What do you notice about that sum in relation to the triangle?

Answers to homework (a) ∠𝑎=73° (b) ∠𝑏=49° (c) ∠𝑐=58° (d) ∠𝑎+∠𝑏+∠𝑐=73°+49°+58°=180°. The interior angles of the triangle add up to 180°.

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