From First Triangle to the Final Proof
This workbook builds, one short step at a time, every idea you need to solve the capstone problem at the end — a congruence puzzle that asks you to find the ratio AB⁄BC using nothing but the definition of a triangle, a midsegment, and CPCTC. Read the Concept and Worked Example in each unit before you try the practice questions. Answer, check, and read the explanation — win or lose — then move to the next unit.
The Parts of a Triangle
A triangle has exactly three vertices (corner points), three sides (the segments joining the vertices), and three angles (formed at each vertex). We name a triangle by listing its vertices, for example △PQR. Its sides are PQ, QR, and RP; its angles are ∠P, ∠Q, and ∠R.
Q: In △PQR above, name the side opposite vertex Q.
A: The side "opposite" a vertex is the one that does not touch it. The side that does not
touch Q is PR.
In △RST, which of the following is a side of the triangle?
How many angles does every triangle have?
Classifying Triangles by Their Sides
Scalene: no sides equal. Isosceles: exactly two sides equal. Equilateral: all three sides equal. Diagrams mark equal sides with matching tick marks (small hash lines) — sides with the same number of ticks are equal in length.
Q: A triangle has sides of length 6, 6, and 9. What type is it?
A: Two sides (6 and 6) are equal, so it is isosceles.
A triangle has sides 5, 5, and 8. What type of triangle is it?
If a triangle's three sides all carry the same number of tick marks, it is called a(n):
Base Angles of an Isosceles Triangle
In an isosceles triangle, the two equal sides are called legs, and the third side is the base. The two angles touching the base (the base angles) are always equal to each other.
Q: In △ABC, AB = AC and ∠B = 62°. Find ∠C.
A: AB = AC means the legs are AB and AC, so the base is BC, and the base angles are
∠B and ∠C. Base angles are equal, so ∠C = 62°.
In isosceles △ABC with AB = AC, which two angles must be equal?
In △ABC, AB = AC and ∠B = 50°. Find ∠C (in degrees). Enter a number only.
Midpoints
The midpoint of a segment is the point exactly halfway between its two endpoints. If D is the midpoint of AB, then AD = DB = ½AB.
Q: D is the midpoint of AB, and AB = 14. Find AD.
A: AD = ½AB = ½(14) = 7.
D is the midpoint of segment AB, and AB = 18. What is AD?
If AD = DB, then D is called the ____ of AB.
The Triangle Midsegment Theorem
If D and E are the midpoints of two sides of a triangle (say AB and AC), the segment DE is called a midsegment. The Midsegment Theorem says: DE ∥ BC and DE = ½BC.
Q: D, E are midpoints of AB, AC. If BC = 18, find DE.
A: DE = ½BC = ½(18) = 9.
In △ABC, D and E are midpoints of AB and AC. If BC = 20, find DE.
Which statement correctly describes midsegment DE (D, E midpoints of AB, AC)?
Congruence Statements & Correspondence Order
Two triangles are congruent (≅) if they are exactly the same size and shape. The order of the letters in a congruence statement tells you exactly which vertices match up. If △ABC ≅ △XYZ, then A↔X, B↔Y, C↔Z — matched in the order they're written, not by position in a picture.
Q: △ABC ≅ △DEF. Which side corresponds to AB?
A: A↔D and B↔E, so AB corresponds to DE.
If △PQR ≅ △STU, which vertex corresponds to Q?
If △PQR ≅ △STU, which side corresponds to QR?
Corresponding Parts of Congruent Triangles
CPCTC stands for "Corresponding Parts of Congruent Triangles are Congruent." Once you know two triangles are congruent, every pair of corresponding sides and corresponding angles must be equal — not just the ones you started with.
Q: △ABC ≅ △DEF. List every side equality given by CPCTC.
A: A↔D, B↔E, C↔F, so AB = DE, BC = EF, and CA = FD.
Given △XYZ ≅ △LMN, which of these is TRUE by CPCTC?
△ABC ≅ △PQR. If AB = 8, what is PQ?
Turning Correspondence into Equations
When two triangles are congruent, you can set corresponding sides equal — even when one is written with a variable — and then solve the resulting equation for the unknown.
Q: △ABC ≅ △DEF. AB = 2x and DE = 10. Find x.
A: By CPCTC, AB = DE, so 2x = 10 → x = 5.
△ABC ≅ △DEF. BC = 3x and EF = 15. Find x.
△ABC ≅ △DEF. AB = x + 2 and DE = 9. Find x.
When a Triangle Is Congruent to Itself, Reordered
Sometimes a congruence statement reuses the same three points in a different order, like △ABC ≅ △CAB. Read the correspondence exactly the same way as before — letter by letter — and it reveals new equal sides hiding inside a single triangle.
Q: △ABC ≅ △CAB. What can you conclude about △ABC?
A: Matching letter by letter: A↔C, B↔A, C↔B. So AB↔CA, BC↔AB, and
CA↔CB. That gives AB = CA, BC = AB, and CA = CB — all three sides are equal, so △ABC is
equilateral.
△ABC ≅ △BCA. What can you conclude about △ABC?
△ABC ≅ △BAC. What can you conclude about △ABC?
The Original Problem
In △ABC, D and E are the midpoints of AB and AC. Line DE is extended to a point F. It is given that △ABC ≅ △BFA. Find AB ⁄ BC.
Need a nudge? Open hints one at a time
- Write out the CPCTC equalities from △ABC ≅ △BFA (A↔B, B↔F, C↔A). What three side-equalities do you get?
- One of those equalities forces △ABC to be isosceles. Which sides turn out equal?
- D and E are midpoints, so DE ∥ BC (Unit 05) — and F sits on that same line. Combine this with the CPCTC side lengths and set up variables for AB and BC to solve for the ratio.
Find AB ⁄ BC.
Show the full worked solution
Let B = (0, 0), C = (a, 0), and since AB = AC = c, A = (a⁄2, h) with h² = c² − a²⁄4. The midpoints are D = (a⁄4, h⁄2) and E = (3a⁄4, h⁄2), so line DE is the horizontal line y = h⁄2, and F = (x, h⁄2) for some x.
From CPCTC, BF = c and FA = a, which give two distance equations:
x² + (h⁄2)² = c² and (x − a⁄2)² + (h⁄2)² = a²
Subtracting eliminates (h⁄2)² and gives x in terms of a and c. Substituting back into the first equation and letting k = c⁄a produces the quadratic k⁴ − 2.25k² + 0.5 = 0 in u = k², whose roots are u = 2 or u = 0.25. The root u = 0.25 (k = 0.5) makes AB + AC exactly equal to BC, which is a degenerate (flat) triangle, so it is rejected. The valid answer is u = 2, so k = AB⁄BC = √2.
Three More Problems for Practice
Same tools, fresh numbers — CPCTC, isosceles triangles, and the midsegment theorem.
△ABC ≅ △DEF. AB = 3x − 1 and DE = 11. Find x.
In △ABC, D and E are midpoints of AB and AC. If BC = 16, find DE.
△ABC ≅ △BCA and AB = 5. Find BC.