SAT-69: Isosceles and Equilateral Triangles

SAT-69: Isosceles and Equilateral Triangles

Description: Special triangles have special angle rules. An isosceles triangle has two equal sides and two equal base angles; an equilateral triangle has all sides equal and all angles 60°. These facts, combined with the 180° sum, solve many SAT figures quickly.

Isosceles triangle

An isosceles triangle has (at least) two equal sides. The angles opposite those equal sides — the base angles — are also equal. So if you know one base angle, you know the other. (Oʻzbekcha: teng yonli uchburchakda teng tomonlar qarshisidagi burchaklar ham teng.)

Equilateral triangle

An equilateral triangle has all three sides equal, so all three angles are equal. Since they sum to 180°, each is 180 ÷ 3 = 60°. (Oʻzbekcha: teng tomonli uchburchakda har bir burchak 60° ga teng.)

The two-way connection (sides ↔ angles)

Equal sides force equal angles, and equal angles force equal sides. So you can reason in either direction: spotting two equal angles tells you two sides are equal, which can unlock a side length. (Oʻzbekcha: teng tomonlar → teng burchaklar va aksincha.)

Worked Example 1 — find base angles

An isosceles triangle has a vertex (top) angle of 40°. Find each base angle.

• The two base angles are equal; call each b. Then 40 + b + b = 180 → 2b = 140 → b = 70° each.

Worked Example 2 — find the vertex angle

An isosceles triangle has base angles of 55° each. Find the vertex angle.

• Vertex = 180 − 55 − 55 = 70°.

(Oʻzbekcha: uchidagi burchak = 180 − ikkita asos burchagi.)

Worked Example 3 — equilateral with algebra

A triangle has all sides equal, and one angle is labelled (5x)°. Find x.

• Equilateral → every angle is 60°, so 5x = 60 → x = 12.

Tip: tick marks on a figure show equal sides; equal sides mean equal opposite angles. Always read the tick marks. (Oʻzbekcha: rasmda teng tomon belgisi (chiziqcha) boʻlsa, ularning qarshisidagi burchaklar teng.)

Worked Example 4 — use equal angles to find a side

In triangle ABC, angle B = angle C = 65°, and side AB = 9. What can you say about side AC?

• Equal angles force the opposite sides to be equal. Angle B is opposite AC and angle C is opposite AB.
• Since angle B = angle C, the sides opposite them are equal: AC = AB = 9.

This is the "reason backwards" direction — from equal angles to equal sides. (Oʻzbekcha: teng burchaklardan teng tomonlarga xulosa chiqaramiz.)

A useful special case

An equilateral triangle is also isosceles (all the rules still apply), and because every angle is 60°, it is the only triangle that is also equiangular. On the SAT, spotting "all sides equal" instantly gives you three 60° angles — and spotting "all angles 60°" instantly gives you three equal sides. Either clue unlocks the whole triangle. (Oʻzbekcha: teng tomonli uchburchak ham teng yonli; barcha burchaklari 60° boʻlgani uchun tomonlar ham teng.)

Common mistake

Don't assume a triangle is isosceles just because it looks like it. You need either equal tick marks on the sides or two equal angles stated. Without one of those, the base-angle rule does not apply. (Oʻzbekcha: faqat koʻrinishiga qarab teng yonli deb hisoblamang — belgisi yoki teng burchaklari boʻlishi kerak.)

Practice 1

An isosceles triangle has a base angle of 48°. Find the vertex angle.

Practice 2

In an isosceles triangle, the vertex angle is (2x)° and each base angle is (x + 30)°. Find x.

Key words — Kalit soʻzlar

• Isosceles triangle — teng yonli uchburchak
• Equilateral triangle — teng tomonli uchburchak
• Base angles — asos burchaklari
• Vertex angle — uchidagi burchak
• Equal sides — teng tomonlar
• Opposite — qarshi (roʻparadagi)
• Tick marks — tenglik belgilari
• Congruent — teng (mos)
• Angle sum — burchaklar yigʻindisi

Summary

• Isosceles: two equal sides → two equal base angles.
• Equilateral: all sides equal → every angle is 60°.
• Equal sides ↔ equal opposite angles (reason both ways), then use the 180° sum.