Proof of angle-angle similarity with the law of sines

Proof

We get the third angle from the angle sum, and then the ratio of the sides from the law of sines.

Let $\alpha = \angle BAC$ , $\beta = \angle CBA$ and $\gamma = \angle ACB$ , and let $\alpha'$ , $\beta'$ , $\gamma'$ be the corresponding angles of $A'B'C'$ . By hypothesis $\alpha = \alpha'$ and $\beta = \beta'$ .
By the angle sum of a triangle, $\gamma = \pi - \alpha - \beta = \pi - \alpha' - \beta' = \gamma'$ .
Let $a = d(B, C)$ , $b = d(C, A)$ and $c = d(A, B)$ , with the Euclidean distance $d$ . Let $a'$ , $b'$ , $c'$ be the same distances in $A'B'C'$ .
By the law of sines in $ABC$ , there is a number $k$ with $a = k \sin \alpha$ , $b = k \sin \beta$ and $c = k \sin \gamma$ . Here $k = a / \sin \alpha$ .
By the law of sines in $A'B'C'$ and steps 1 and 2, there is a number $k'$ with $a' = k' \sin \alpha$ , $b' = k' \sin \beta$ and $c' = k' \sin \gamma$ .
The vertices of a triangle are distinct, so $a, a' > 0$ . The quotient $a / \sin \alpha$ in the law of sines is defined, so $\sin \alpha \neq 0$ . Since $\alpha \in [0, \pi]$ , where the sine is nonnegative, $\sin \alpha > 0$ . So $k > 0$ and $k' = a' / \sin \alpha > 0$ .
Let $r = k' / k > 0$ . By steps 4 and 5, $a' = r a$ , $b' = r b$ and $c' = r c$ .
This says $d(B', C') = r \, d(B, C)$ , $d(C', A') = r \, d(C, A)$ and $d(A', B') = r \, d(A, B)$ .
Let $A'' = A' / r$ , $B'' = B' / r$ and $C'' = C' / r$ . Scaling by $1 / r$ keeps points non-collinear, so $A''B''C''$ is a triangle.
Scaling by $1 / r$ divides each distance by $r$ . By step 8, $A''B''C''$ has the same side lengths as $ABC$ .
By side-side-side congruence, there is an isometry $f$ of $\R^n$ with $f(A) = A''$ , $f(B) = B''$ and $f(C) = C''$ .
Let $g(x) = r f(x)$ . Then $g$ maps $A$ , $B$ , $C$ to $A'$ , $B'$ , $C'$ , and $d(g(x), g(y)) = r \, d(x, y)$ for all $x, y$ .
So $A'B'C'$ is a copy of $ABC$ , scaled by $r$ after a rigid motion. Hence the triangles are similar with the ratio $r$ .

Idea

Two angles fix the third, because the three add up to $\pi$ . The law of sines then says that the sides are proportional to the sines of the opposite angles. Equal angles give equal sines, so the sides of the two triangles are proportional with one common factor. Side-side-side congruence then turns this into a scaling after a rigid motion.

Reading path

To understand this page from the ground up, read these 323 pages in order. The foundations come first. The path takes the proofs Proof of the Steinitz exchange lemma by induction, Proof of the Cauchy–Hadamard theorem, Proof of the derivative of cosine by term-by-term differentiation, Proof of the Weierstrass M-test, Proof of the limit of a sum of sequences, Proof of Rolle's theorem, Proof of the approximation property of the supremum, Proof that every sequence has a monotone subsequence, by peaks, Proof of the formula for the derivative of a polynomial, Proof of the commutativity of addition by induction, Proof of term-by-term differentiation, Proof of the epsilon-delta form of continuity, Proof of the power rule by induction, Proof that absolutely convergent series converge, Proof of the sum of a finite geometric series, Proof that convergent sequences are Cauchy, Proof that a complex number times its conjugate is the square of its modulus, Proof of the existence of an orthonormal basis by the Gram–Schmidt process, Proof of the Cauchy–Schwarz inequality, Proof that there is a natural number n with 1/n below any positive epsilon, Proof of the chain rule with an auxiliary difference quotient, Proof of side-side-side congruence with orthonormal frames, Proof of the Cauchy criterion for series, Proof of the Cauchy criterion for sequences by the Bolzano–Weierstrass theorem, Proof that cos pi = -1 and sin pi = 0, Proof of the equality of complex numbers, Proof that the reals form an ordered field, with the Cauchy sequence construction, Proof of the root test, Proof of the double negation law by a truth table, Proof that power series converge uniformly on smaller intervals inside the radius of convergence, Proof of the extreme value theorem by the Bolzano–Weierstrass theorem, Proof of the uniqueness of limits, Proof of strong induction from ordinary induction, Proof that power series are differentiable inside the radius of convergence, Proof of the limit of a product of sequences, Proof that uniform convergence of derivatives allows term-by-term differentiation, Proof that the absolute value is multiplicative, Proof of the squeeze theorem for sequences, Proof of the existence of square roots of nonnegative reals, Proof of Pascal's rule from the recursive definition, Proof that the Gram–Schmidt vectors are orthogonal, Proof of the linearity of differentiation, Proof of the angle addition formulas by differentiation, Proof of the angle sum of a triangle with sines and cosines, Proof of the sequential characterization of limits, Proof that Cauchy sequences are bounded, Proof that an orthonormal family is linearly independent, Proof of the triangle inequality for real numbers, Proof of the epsilon-N definition of the limit of a sequence, Proof of the law of sines from the law of cosines, Proof of the comparison test, Proof of Fermat's theorem on stationary points, Proof that limits preserve non-strict inequalities, Proof that every linearly independent set extends to a basis, by Zorn's lemma, Proof of the factorial formula for binomial coefficients by induction, Proof that a function with positive derivative is strictly increasing, Proof of Zorn's lemma, Proof of the intermediate value theorem by the supremum, Proof of the binomial theorem by induction, Proof that exponentials dominate powers in sequences, by the ratio of consecutive terms, Proof that differentiable functions are continuous, Proof of the epsilon-delta form of the limit of a function, Proof of the derivative of sine by term-by-term differentiation, Proof of the well-ordering principle from strong induction, Proof that n independent or n spanning vectors in an n-dimensional space form a basis, Proof of the Bolzano–Weierstrass theorem by a monotone subsequence, Proof of the Archimedean property in an Archimedean ordered field, Proof that a composition of continuous functions is continuous, Proof of the Pythagorean trigonometric identity by differentiation, Proof that a decreasing sequence bounded below converges, Proof of the term test, Proof that the reals are Archimedean from the completeness axiom, Proof of the ratio test, Proof that convergent sequences are bounded, Proof of the law of cosines with the dot product, Proof of the product rule, Proof of the monotone convergence theorem for sequences, Proof that continuity is equivalent to sequential continuity, Proof that a function with zero derivative on an open interval is constant, Proof of the boundedness theorem by Bolzano–Weierstrass, Proof that a series of nonnegative terms converges iff its partial sums are bounded, Proof of the mean value theorem from Rolle's theorem, Proof that the Cauchy product of absolutely convergent series converges absolutely to the product, Proof of the constant multiple rule.

Angle at a point concept
Arccosine concept
Binomial coefficient definition
Collinear points concept
Complex numbers object
Complex numbers as pairs of reals concept
Congruent triangles concept
Construction of the reals from Cauchy sequences concept
Definition by recursion concept
Dot product concept
Element definition
Euclidean distance concept
Factorial definition
Group concept
Imaginary unit object
Inner product concept
Inner product space concept
Integers object
Isometry concept
Length of a vector concept
Limit point of a set definition
Linear independence concept
Linear map concept
Local maximum definition
Mathematical induction concept
Natural numbers object
Norm induced by an inner product concept
Ordered field definition
Orthonormal basis concept
Orthonormal family concept
Peano axioms concept
Proof of Zorn's lemma proof
Proposition concept
Rational numbers object
Real numbers object
Sequence definition
Set bounded above definition
Similar triangles concept
Sine and cosine concept
Strictly monotone function definition
Triangle concept
Vector space definition
Proof of Pascal's rule from the recursive definition proof
Real and imaginary parts definition
Set definition
Ring definition
Proof of the law of cosines with the dot product proof
Ring of integers statement
Limit of a function definition
Local minimum definition
Archimedean ordered field definition
Successor function definition
Conjunction definition
Disjunction definition
Implication definition
Negation definition
Predicate definition
The reals form a field statement
Real sequence definition
Subsequence definition
Infimum definition
Supremum definition
Inner product definition
Linear combination definition
Linear subspace definition
Norm definition
Pascal's rule statement
Family of sets definition
Ordered pair definition
Complex conjugate definition
Commutative ring definition
Continuity at a point definition
Derivative definition
One-sided limit definition
Union definition
Biconditional definition
Truth table concept
Modus ponens concept
Law of cosines statement
Addition of natural numbers definition
Existential quantifier definition
Universal quantifier definition
Bounded sequence definition
Cauchy sequence definition
Geometric sequence definition
Limit of a sequence definition
Monotone sequence definition
Sequence tending to infinity definition
Series definition
Proof of the Archimedean property in an Archimedean ordered field proof
The rationals form a field statement
The successor function is injective statement
Zero is not a successor statement
Inner product space definition
Orthogonal vectors definition
Linear independence definition
Linear span definition
Completeness axiom statement
Cartesian product definition
Proof of the factorial formula for binomial coefficients by induction proof
Continuous function definition
Proof of the equality of complex numbers proof
Differentiable function definition
One-sided derivative definition
Union of a family of sets definition
Subset definition
Mathematical proof concept
Propositional logic concept
Uniqueness quantifier definition
Proof of the commutativity of addition by induction proof
Archimedean property statement
Limit superior definition
Proof of the sum of a finite geometric series proof
Convergent sequence definition
Order on the natural numbers definition
Pointwise convergence definition
Sequential continuity definition
The sequence r^n tends to infinity for r > 1 example
Cauchy product definition
Partial sum definition
Power series definition
Norm induced by an inner product definition
The rationals form an ordered field statement
Orthonormal family definition
Spanning set definition
Equality of complex numbers statement
Binary relation definition
Proof that a composition of continuous functions is continuous proof
Factorial formula for binomial coefficients statement
Tautology definition
Order on the reals definition
Commutativity of addition statement
Sum of a finite geometric series statement
Uniform convergence definition
Proof of strong induction from ordinary induction proof
Radius of convergence definition
Convergent series definition
Sine and cosine definition
Proof that an orthonormal family is linearly independent proof
Finite-dimensional vector space definition
Proof of the Steinitz exchange lemma by induction proof
Basis definition
Gram–Schmidt process concept
Antisymmetric relation definition
Reflexive relation definition
Transitive relation definition
Proof of the binomial theorem by induction proof
Interval definition
Function definition
Logical equivalence definition
Uniform convergence on a set definition
Dimension definition
Series of functions definition
Dimension theorem for vector spaces statement
Sum of a series definition
Strong induction statement
Absolute convergence definition
Orthonormal basis definition
An orthonormal family is linearly independent statement
Steinitz exchange lemma statement
Proof that the Gram–Schmidt vectors are orthogonal proof
Preorder definition
Binomial theorem statement
Proof of the epsilon-delta form of continuity proof
Proof of the epsilon-delta form of the limit of a function proof
Choice function definition
Proof of the well-ordering principle from strong induction proof
Proof of the double negation law by a truth table proof
Gram–Schmidt vectors are orthogonal statement
Proof that n independent or n spanning vectors in an n-dimensional space form a basis proof
Partial order definition
Epsilon-delta definition of continuity statement
Axiom of choice statement
Epsilon-delta definition of a limit statement
Double negation law statement
Well-ordering principle statement
Linearly independent set of the right size is a basis statement
Maximal element definition
Total order definition
Upper bound definition
Composition of continuous functions is continuous statement
Proof by contradiction concept
Proof that every sequence has a monotone subsequence, by peaks proof
Chain definition
Proof that the reals form an ordered field, with the Cauchy sequence construction proof
Every sequence has a monotone subsequence statement
Zorn's lemma statement
The reals form an ordered field statement
Absolute value definition
Proof that every linearly independent set extends to a basis, by Zorn's lemma proof
Proof of the approximation property of the supremum proof
Proof of the existence of square roots of nonnegative reals proof
Proof of the triangle inequality for real numbers proof
Proof that the absolute value is multiplicative proof
Proof of the epsilon-N definition of the limit of a sequence proof
Every linearly independent set extends to a basis statement
Existence of square roots of nonnegative reals statement
Approximation property of the supremum statement
Triangle inequality statement
Absolute value is multiplicative statement
Square root definition
Epsilon-N definition of the limit of a sequence statement
Proof of the existence of an orthonormal basis by the Gram–Schmidt process proof
Proof of the intermediate value theorem by the supremum proof
Proof that the reals are Archimedean from the completeness axiom proof
Proof of the limit of a sum of sequences proof
Proof of the squeeze theorem for sequences proof
Modulus of a complex number definition
Proof of the uniqueness of limits proof
Proof that limits preserve non-strict inequalities proof
Proof that differentiable functions are continuous proof
Intermediate value theorem statement
Proof that convergent sequences are bounded proof
Proof that convergent sequences are Cauchy proof
Proof that Cauchy sequences are bounded proof
The sequence 1/n tends to zero example
The reals are Archimedean statement
Existence of an orthonormal basis statement
Proof of the term test proof
Proof of the monotone convergence theorem for sequences proof
Proof that a complex number times its conjugate is the square of its modulus proof
Differentiable functions are continuous statement
Limit of a sum of sequences statement
Uniqueness of limits statement
Limits preserve non-strict inequalities statement
Squeeze theorem statement
Convergent sequences are bounded statement
Proof that there is a natural number n with 1/n below any positive epsilon proof
Convergent sequences are Cauchy statement
Cauchy sequences are bounded statement
Monotone convergence theorem for sequences statement
Proof of side-side-side congruence with orthonormal frames proof
A complex number times its conjugate statement
Proof of the limit of a product of sequences proof
There is a natural number n with 1/n below any positive epsilon statement
Side-side-side congruence statement
Term test statement
Proof that a series of nonnegative terms converges iff its partial sums are bounded proof
The sequence r^n tends to zero for |r| < 1 example
Proof that a decreasing sequence bounded below converges proof
Proof of the sequential characterization of limits proof
Proof that continuity is equivalent to sequential continuity proof
Decreasing sequence bounded below converges statement
Limit of a product of sequences statement
Geometric series example
A series of nonnegative terms converges iff its partial sums are bounded statement
Proof of the Cauchy–Schwarz inequality proof
Continuity is equivalent to sequential continuity statement
Proof that the Cauchy product of absolutely convergent series converges absolutely to the product proof
Sequential characterization of limits statement
Proof of the root test proof
Proof of the ratio test proof
Proof of the Cauchy–Hadamard theorem proof
Proof of the Bolzano–Weierstrass theorem by a monotone subsequence proof
Proof that exponentials dominate powers in sequences, by the ratio of consecutive terms proof
Proof of the comparison test proof
Cauchy–Schwarz inequality statement
Proof of the chain rule with an auxiliary difference quotient proof
Proof of the product rule proof
Proof of the linearity of differentiation proof
Comparison test statement
Bolzano–Weierstrass theorem statement
Cauchy product of absolutely convergent series statement
Exponentials dominate powers in sequences statement
Product rule statement
Chain rule statement
Linearity of differentiation statement
Ratio test statement
Root test statement
Proof of the boundedness theorem by Bolzano–Weierstrass proof
Proof of the Cauchy criterion for sequences by the Bolzano–Weierstrass theorem proof
Proof of the constant multiple rule proof
Proof of the power rule by induction proof
Boundedness theorem statement
Cauchy–Hadamard theorem statement
Cauchy criterion for sequences statement
Constant multiple rule statement
Power rule statement
Proof of the extreme value theorem by the Bolzano–Weierstrass theorem proof
Uniformly Cauchy sequence of functions definition
Proof of the Cauchy criterion for series proof
Extreme value theorem statement
Proof of the formula for the derivative of a polynomial proof
Proof of Fermat's theorem on stationary points proof
Cauchy criterion for series statement
Derivative of a polynomial statement
Proof that absolutely convergent series converge proof
Fermat's theorem on stationary points statement
Absolutely convergent series converge statement
Proof of Rolle's theorem proof
Rolle's theorem statement
Proof of the Weierstrass M-test proof
Proof of the mean value theorem from Rolle's theorem proof
Weierstrass M-test statement
Mean value theorem statement
Proof that power series converge uniformly on smaller intervals inside the radius of convergence proof
Proof that a function with zero derivative on an open interval is constant proof
Proof that uniform convergence of derivatives allows term-by-term differentiation proof
Power series converge uniformly on smaller discs inside the radius of convergence statement
Proof that a function with positive derivative is strictly increasing proof
A function with zero derivative is constant statement
Uniform convergence of derivatives allows term-by-term differentiation statement
A function with positive derivative is strictly increasing statement
Proof of term-by-term differentiation proof
Term-by-term differentiation statement
Proof that power series are differentiable inside the radius of convergence proof
Power series are differentiable inside the radius of convergence statement
Proof of the derivative of cosine by term-by-term differentiation proof
Proof of the derivative of sine by term-by-term differentiation proof
Derivative of cosine statement
Derivative of sine statement
Proof of the angle addition formulas by differentiation proof
Proof of the Pythagorean trigonometric identity by differentiation proof
pi object
Pythagorean trigonometric identity statement
Angle addition formulas statement
Proof that cos pi = -1 and sin pi = 0 proof
Proof of the law of sines from the law of cosines proof
cos pi = -1 statement
Law of sines statement
Proof of the angle sum of a triangle with sines and cosines proof
Angle sum of a triangle statement
Proof of angle-angle similarity with the law of sines proof

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Angle-angle similarity

Tag	`R9THY4JCVQ`
URI	`m2://R9THY4JCVQ`
Kind	proof
Writing	W0 stub
MSC	51M04
Proves	Angle-angle similarity
Uses	Similar triangles, Angle at a point, Angle sum of a triangle, Law of sines, Euclidean distance, Triangle, Sine and cosine, Side-side-side congruence, Isometry
Changed	4 November 2026

Proof of angle-angle similarity with the law of sines

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