Question

Let A = (1,4), B = (0,−9), C = (7,2), and D = (6,9). Prove that angles DAB and DCB are the same size. Can anything be said about the angles ABC and ADC?

Answer #1

Let A = (0,0), B = (8,1), C = (5,−5), P = (0,3), Q = (7,7), and
R = (1,10). Prove that angles ABC and PQR have the same size.

Let A = (2, 9), B = (6, 2), and C = (10, 10). Verify that
segments AB and AC have the same length. Measure angles ABC and
ACB. On the basis of your work, propose a general statement that
applies to any triangle that has two sides of equal length. Prove
your assertion, which might be called the Isosceles-Triangle
Theorem.

Let A, B, C and D be sets. Prove that A\B ⊆ C \D if and only if
A ⊆ B ∪C and A∩D ⊆ B

Let A, B, C and D be sets. Prove that A \ B and C \ D are
disjoint if and only if A ∩ C ⊆ B ∪ D.

Prove if 0 < a < b and 0 < c < d then 0 < ac <
bd in two a two-column proof format.

Let a, b, c, m be integers with m > 0. Prove the following:
(a) ”a ≡ 0 (mod 2) if and only if a is even” and ”a ≡ 1 (mod 2) if
and only if a is odd”. (b) a ≡ b (mod m) if and only if a − b ≡ 0
(mod m) (c) a ≡ b (mod m) if and only if (a mod m) = (b mod m).
Recall from Definition 8.10 that (a...

1. Suppose that a = d · k + b, where a, b, d, k are all
integers. Prove that b is divisible by d if and only if a is
divisible by d.
Let x = abc be a three-digit number with digits a, b, c (so a,
b, c ∈ {0, 1, 2, . . . 9}). Prove that x is divisible by 3 if and
only if a + b + c is divisible by 3.

9. Let a, b, q be positive integers, and r be an integer with 0
≤ r < b. (a) Explain why gcd(a, b) = gcd(b, a). (b) Prove that
gcd(a, 0) = a. (c) Prove that if a = bq + r, then gcd(a, b) =
gcd(b, r).

(a) Let a,b,c be elements of a field F. Prove that if a not= 0,
then the equation ax+b=c has a unique solution.
(b) If R is a commutative ring and x1,x2,...,xn are independent
variables over R, prove that R[x σ(1),x σ (2),...,x σ (n)] is
isomorphic to R[x1,x2,...,xn] for any permutation σ of the set
{1,2,...,n}

Let O ∈ (AB) and C /∈ AB. Prove that there is a point D on the
same side of AB as C such that m(∠DOA) = m(∠COB).

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