SAT-68: Triangles — Interior and Exterior Angle Theorems

SAT-68: Triangles — Interior and Exterior Angle Theorems

Description: Two theorems unlock most triangle-angle problems: the three interior angles always sum to 180°, and an exterior angle equals the sum of the two non-adjacent interior angles. Together they let you find any missing angle.

Interior Angle Theorem

The three interior angles of any triangle add up to 180°.

This holds for every triangle, no matter its shape. (Oʻzbekcha: istalgan uchburchakning uchta ichki burchagi yigʻindisi 180° ga teng.)

Exterior Angle Theorem

If you extend one side of a triangle, the exterior angle formed equals the sum of the two interior angles not next to it (the "remote" interior angles). It also makes a straight line with its own interior angle, so they are supplementary. (Oʻzbekcha: tashqi burchak unga qoʻshni boʻlmagan ikki ichki burchak yigʻindisiga teng.)

Worked Example 1 — find the third angle

A triangle has angles 50° and 70°. Find the third.

• Sum is 180: third = 180 − 50 − 70 = 60°.

Worked Example 2 — solve with variables

A triangle's angles are x, 2x, and 3x. Find each angle.

• x + 2x + 3x = 180 → 6x = 180 → x = 30.
• Angles: 30°, 60°, 90° (a right triangle).

(Oʻzbekcha: barcha burchaklarni qoʻshib 180° ga tenglaymiz.)

Worked Example 3 — exterior angle theorem

An exterior angle of a triangle measures 120°. The two remote interior angles are equal. Find each.

• Exterior = sum of the two remote interior angles: 120 = a + a = 2a → a = 60° each.
• Check: the adjacent interior angle is 180 − 120 = 60°, and 60 + 60 + 60 = 180 ✓.

Shortcut: the exterior angle gives you the sum of the two far angles instantly — no need to find the adjacent angle first. (Oʻzbekcha: tashqi burchak ikki uzoq burchak yigʻindisini darhol beradi.)

Worked Example 4 — combine with isosceles facts

A triangle has two equal sides and the angle between the two equal sides (the vertex) is 80°. Find the other two angles.

• Equal sides mean the two base angles are equal; call each b.
• Angle sum: 80 + b + b = 180 → 2b = 100 → b = 50° each.
• So the angles are 80°, 50°, 50° (and 80 + 50 + 50 = 180 ✓).

This shows how the 180° rule teams up with side facts you will study next in SAT-69. (Oʻzbekcha: 180° qoidasi teng tomon faktlari bilan birga ishlaydi.)

Why the exterior-angle theorem is true

The exterior angle and its neighbouring interior angle form a straight line, so they add to 180°. But the three interior angles also add to 180°. Subtracting the shared interior angle from both statements leaves: exterior angle = the other two interior angles. That is the whole proof in one line — and it is why the theorem always holds. (Oʻzbekcha: tashqi burchak qoʻshni ichki burchak bilan 180° hosil qiladi; uchta ichki burchak ham 180°, shuning uchun tashqi = qolgan ikki ichki burchak.)

Practice 1

A right triangle has one angle of 35° (besides the 90°). Find the third angle.

Practice 2

An exterior angle of a triangle is (2x + 10)° and the two remote interior angles are 40° and 70°. Find x.

Key words — Kalit soʻzlar

• Triangle — uchburchak
• Interior angle — ichki burchak
• Exterior angle — tashqi burchak
• Remote interior angles — uzoq (qoʻshni boʻlmagan) ichki burchaklar
• Sum — yigʻindi
• Right triangle — toʻgʻri burchakli uchburchak
• Extend a side — tomonni davom ettirish
• Adjacent — qoʻshni
• Theorem — teorema

Summary

• Interior angles of a triangle sum to 180°.
• An exterior angle = sum of the two remote interior angles.
• Set the relevant angles into one equation and solve.