Geometry Honors Proofs and Circles and Trig

What does CPCTC stand for?

Corresponding Parts of Congruent Triangles are Congruent

When can you use CPCTC in a proof?

After proving two triangles are congruent

What property states that any segment or angle is congruent to itself?

Reflexive Property

What do vertical angles prove in triangle congruence proofs?

They are congruent and can justify angle congruence

What is the definition of a midpoint?

The point that divides a segment into two congruent segments

How does the definition of a bisector help in proofs?

It divides a segment or angle into two congruent parts

What does SSS stand for in triangle congruence?

Side-Side-Side

What does SAS stand for in triangle congruence?

Side-Angle-Side

What does ASA stand for in triangle congruence?

Angle-Side-Angle

What does AAS stand for in triangle congruence?

Angle-Angle-Side

What does HL stand for in triangle congruence?

Hypotenuse-Leg (used only in right triangles)

What must be shown before using HL in a proof?

That both triangles are right triangles and have congruent hypotenuse and one leg

What's the first step in most two-column triangle proofs?

State the given information

What reason is used when a side is shared between two triangles?

Reflexive Property

How do you prove two lines are parallel in a proof?

Show alternate interior angles or corresponding angles are congruent

Why is proving triangles congruent important?

So you can use CPCTC to prove parts are congruent

What is the Transitive Property?

If a = b and b = c, then a = c

What is the Substitution Property?

If a = b, you can replace b with a in an expression

What's the difference between a postulate and a theorem?

Postulates are assumed true without proof; theorems are proven

How do you prove a segment is a midpoint in a proof?

Show that it divides a segment into two congruent parts

How do you prove a quadrilateral is a parallelogram?

Show both pairs of opposite sides are congruent or parallel

How do you prove a parallelogram is a rectangle?

Show it's a parallelogram with one right angle or diagonals congruent

How do you prove a parallelogram is a rhombus?

Show it's a parallelogram with perpendicular diagonals or all sides congruent

What is the definition of a kite in a proof?

A quadrilateral with two distinct pairs of adjacent congruent sides

How do you prove two triangles are similar?

Use AA, SAS, or SSS similarity postulates

What is the Triangle Proportionality Theorem?

If a line is parallel to one side of a triangle, it divides the other two sides proportionally

How is similarity used in proofs?

It lets you set up proportions between sides

What is the Converse of the Pythagorean Theorem?

If a² + b² = c², then the triangle is a right triangle

What must be shown to use the Law of Sines in a triangle proof?

The triangle is not a right triangle and you have an angle-opposite side pair

How can the Law of Cosines be used in a proof?

To find missing parts when you know two sides and included angle or all three sides

What must you show to prove two arcs are congruent in a circle proof?

Their central angles are congruent

How do you prove two chords in a circle are congruent?

Show they are equidistant from the center

What is the definition of a tangent line in a proof?

A line that touches a circle at exactly one point

How do you prove a line is tangent to a circle?

Show it is perpendicular to the radius at the point of contact

What is the Radian Arc Length Formula used in proofs?

s = rθ, where θ is in radians

What is the key to solving volume proofs involving Cavalieri's Principle?

Show that two 3D figures have equal heights and equal cross-sectional areas at every level

How do you prove figures have the same volume using dissection?

Break both into congruent parts and show matching volume for each

How do you use conditional probability in a proof?

Show P(A|B) = P(A and B)/P(B), with all values defined

What's the proof strategy for maximizing area or volume?

Show derivative is zero at maximum (if using calculus) or reason geometrically with formulas

What's a key fact about the intersection of medians, altitudes, or angle bisectors in triangle proofs?

They intersect at a point called the centroid, orthocenter, or incenter respectively

How do you prove an angle bisector in a triangle divides the opposite side proportionally?

Use the Angle Bisector Theorem

How do you prove a segment is a perpendicular bisector?

Show it is perpendicular to a segment and divides it into two equal parts

How do you prove a triangle is isosceles?

Show two sides or two angles are congruent

How do you prove a triangle is equilateral?

Show all three sides or all three angles are congruent

What must you show to use the definition of a circle in a proof?

All points on the figure are equidistant from a center point

How do you prove a segment is a diameter of a circle?

Show it passes through the center and touches both sides of the circle

What do you show to prove two polygons are similar?

All corresponding angles are congruent and corresponding sides are proportional

How do you prove that angles in a polygon add to (n-2) × 180°?

Use induction or divide into (n-2) triangles

How do you prove the midpoint formula?

Use the average of x-coordinates and y-coordinates of endpoints

How do you prove the distance formula?

Apply the Pythagorean Theorem on a coordinate plane

How do you prove a triangle is a right triangle using coordinates?

Show that two sides are perpendicular using slope

How do you prove a quadrilateral is a parallelogram using coordinates?

Show both pairs of opposite sides are parallel or congruent using slope/distance

How do you prove a rectangle using coordinates?

Show parallelogram and one right angle (perpendicular sides)

How do you prove a rhombus using coordinates?

Show all sides are congruent using distance formula

How do you prove a square using coordinates?

Show it is a rhombus with one right angle or a rectangle with all sides congruent

How do you prove a trapezoid using coordinates?

Show only one pair of opposite sides are parallel

How do you prove an isosceles trapezoid using coordinates?

Show one pair of opposite sides are parallel and non-parallel sides are congruent

How do you prove a segment is an altitude?

Show it forms a right angle with the opposite side (perpendicular)

How do you prove a segment is a median?

Show it connects a vertex to the midpoint of the opposite side

What is a central angle in a circle?

An angle whose vertex is the center of the circle

What is an inscribed angle?

An angle formed by two chords that have a common endpoint

What is the measure of an inscribed angle?

Half the measure of the intercepted arc

How do you prove two inscribed angles are congruent?

Show they intercept the same arc

What is the formula for the length of an arc (degrees)?

(θ/360) × 2πr

How do you prove a line is a diameter using an inscribed angle?

If an inscribed angle subtends a semicircle, it's a right angle

How do you prove congruent chords in a circle?

They have the same distance from the center

How do you prove segments from the same external point to a circle are equal?

Use the Power of a Point Theorem

What is the Law of Sines?

(sin A)/a = (sin B)/b = (sin C)/c

What is the Law of Cosines?

c² = a² + b² - 2ab cos(C)

How do you prove an angle using inverse trig?

Use sin⁻¹, cos⁻¹, or tan⁻¹ with known side ratios

What is the sine of a complementary angle?

sin(90° - x) = cos(x)

What is the cosine of a complementary angle?

cos(90° - x) = sin(x)

What is SOH-CAH-TOA?

Mnemonic for sine, cosine, and tangent definitions

How do you prove triangle area using trig?

Area = (1/2)ab sin(C)