10.1 Geometry: Angles, Triangles, and Circles

Why Geometry Matters on the USTET

Mathematics is one of four equal-weight USTET subtests (Mental Ability, English, Mathematics, and Science). Within Mathematics, geometry items reward speed: figures are usually given, the relationship is standard, and the distractors are the most common arithmetic mistakes. Grade 11 DepEd geometry — angles, triangles, the Pythagorean theorem, and circle basics — is exactly the depth UST expects. This section trains the relationships you must recall under time pressure; the next section adds perimeter, area, and volume formulas.

Angle Basics and Pair Relationships

An angle is formed by two rays sharing a common endpoint (the vertex). Angles are measured in degrees (°). Classify a single angle by size, then classify pairs by how they relate.

Worked Example 1. Two complementary angles are in the ratio 2 : 7. Find each angle.

Let the angles be 2x and 7x. Then 2x + 7x = 90°, so 9x = 90° and x = 10°. The angles are 20° and 70°.

Worked Example 2. Two lines intersect. One angle measures 3x + 12 and its vertical angle measures 5x − 20. Find x and the angle measure.

Vertical angles are equal: 3x + 12 = 5x − 20 → 12 + 20 = 5x − 3x → 32 = 2x → x = 16. Angle = 3(16) + 12 = 60°.

Parallel Lines Cut by a Transversal

A transversal is a line that intersects two or more other lines. When those lines are parallel, the eight angles that form obey fixed rules:

Practical shortcut: when lines are parallel, every acute angle in the figure equals every other acute angle, every obtuse equals every other obtuse, and any acute plus any obtuse equals 180°. If one labeled angle is 68°, four of the eight angles are 68° and the other four are 112°.

Worked Example 3. Parallel lines m and n are cut by transversal t. An interior angle on one side of t measures 124°. Find (a) its consecutive interior partner and (b) its alternate interior partner.

(a) Consecutive interior: 180° − 124° = 56°. (b) Alternate interior: equal to the given angle, so 124°.

The converse also appears on exams: if corresponding (or alternate interior) angles are congruent, then the lines are parallel. That lets a problem ask "which statement proves the lines are parallel?" rather than "find the angle."

Triangles: Classification and Core Theorems

A triangle is a three-sided polygon. Two classification tables cover nearly every USTET triangle stem.

Angle-sum theorem: the three interior angles of any triangle sum to 180°.

Exterior-angle theorem: an exterior angle equals the sum of the two remote (non-adjacent) interior angles. Equivalent statement: an exterior angle equals 180° minus its adjacent interior angle, which is the same as the sum of the other two.

Triangle inequality: the sum of any two sides must be strictly greater than the third. Sides 4, 7, and 12 cannot form a triangle because 4 + 7 = 11 < 12.

Worked Example 4. In △ABC, ∠A = 52° and ∠B = 67°. Find ∠C.

∠C = 180° − 52° − 67° = 61°.

Worked Example 5. An isosceles triangle has a vertex angle of 48°. Find each base angle.

Base angles are equal: each is (180° − 48°) ÷ 2 = 66°.

Worked Example 6. An exterior angle at vertex C of △ABC measures 110°. If ∠A = 47°, find ∠B.

By the exterior-angle theorem, 110° = ∠A + ∠B = 47° + ∠B, so ∠B = 63°. Check: interior ∠C = 180° − 110° = 70°, and 47° + 63° + 70° = 180°. ✓

Congruence and Similarity (Exam-Ready View)

Two triangles are congruent if corresponding sides and angles match (same size and shape). Common shortcuts tested at Grade 11 depth: SSS, SAS, ASA, AAS, and HL (right triangles only). AAA does not prove congruence — it proves similarity.

Two triangles are similar if corresponding angles are equal and corresponding sides are proportional. Similar triangles let you set up ratios: if △ABC ∼ △DEF with AB corresponding to DE, then AB/DE = BC/EF = AC/DF. USTET items often give two similar right triangles sharing an angle and ask for a missing side via proportion.

The Pythagorean Theorem and Special Right Triangles

In a right triangle, legs a and b and hypotenuse c (opposite the right angle, always longest) satisfy:

a² + b² = c²

Use this only when a right angle is stated or proven. Memorize common Pythagorean triples and their multiples:

45°-45°-90° (isosceles right triangle): legs are equal; hypotenuse = leg × √2. If a leg is x, sides are x : x : x√2.

30°-60°-90°: sides opposite those angles are in the ratio x : x√3 : 2x. Short leg (opposite 30°) = x; long leg (opposite 60°) = x√3; hypotenuse = 2x.

Worked Example 7. Legs 9 cm and 12 cm. Find the hypotenuse.

Recognize 3 × (3-4-5): hypotenuse = 3 × 5 = 15 cm. Or: 81 + 144 = 225 = 15².

Worked Example 8. A 30°-60°-90° triangle has hypotenuse 14. Find both legs.

Hypotenuse = 2x = 14 → x = 7. Short leg = 7; long leg = 7√3.

Worked Example 9. Each leg of an isosceles right triangle is 5. Find the hypotenuse.

Hypotenuse = 5√2. Squaring check: 25 + 25 = 50 = (5√2)². ✓

Circle Essentials and Theorems

Key theorems at USTET depth:

1. Central vs. inscribed: an inscribed angle is half the central angle that subtends the same arc. If a central angle is 80°, an inscribed angle on the same arc is 40°.
2. Angle in a semicircle: an inscribed angle that intercepts a diameter (a semicircle) is a right angle (90°). If AB is a diameter and C is on the circle, ∠ACB = 90°.
3. Inscribed angles on the same arc are congruent.
4. A radius perpendicular to a chord bisects the chord (and the converse in many Grade 11 treatments).
5. Tangent ⊥ radius: a tangent line is perpendicular to the radius drawn to the point of tangency.

Worked Example 10. A central angle measures 100°. An inscribed angle intercepts the same arc. Find the inscribed angle.

Inscribed = half of central = 50°.

Worked Example 11. AB is a diameter of a circle and C is a point on the circle. If ∠BAC = 35°, find ∠ABC.

∠ACB = 90° (angle in a semicircle). Then ∠ABC = 180° − 90° − 35° = 55°.

Common Traps

• Treating vertical angles as supplementary instead of congruent.
• Applying a² + b² = c² to a non-right triangle.
• Confusing corresponding (equal) with consecutive interior (supplementary) under parallel lines.
• Using the full central angle as the inscribed angle instead of taking half.
• Mixing 30°-60°-90°: the side opposite 30° is the short leg (x), not the long leg.

Section Takeaway

Angles: complementary 90°, supplementary 180°, vertical equal, parallel-line pairs as in the table. Triangles: 180° sum, exterior = remote interiors, Pythagorean triples plus 45-45-90 and 30-60-90. Circles: inscribed = ½ arc, angle in a semicircle = 90°. Lock these before moving to area and volume formulas.